class: center, middle, inverse, title-slide # ETC3460 ### Fin ### Semester 1 2018 --- #Part 0 Probability and Statistics * **Features of financial data** - Heavy tails - Asymmetry - Lack of persistence in return levels - Persistent volatility - Volatility Clustering --- * **Transforming non-stationary series to stationary seies** **Simple return** : from time `\(t-1\)` to `\(t\)` Simple gross return `$$1+ R_t =\frac{P_t}{P_{t-1}}$$` Simple net return `$$R_t =\frac{P_t}{P_{t-1}}-1=\frac{P_t-P_{t-1}}{P_{t-1}}$$` -- **Log return** : The natural logarithm of the simple gross return is called the log return `$$r_t=\ln{(1+R_t)}=\ln{\frac{P_t}{P_{t-1}}}=\ln{P_t}-\ln{P_{t-1}}$$` For small `\(R_t\)` , `\(r_t=\ln{(1+R_t)} \approx R_t\)` --- ##Random variable## A __random variable__ is a rule that assigns a numerical outcome to an event in each possible state of the world. (A phenomena that can not be predicted with perfect accuracy) -- * **Sample space** The __sample space__, denoted by `\(\Omega\)` is the set of all possible values that a random variable can take. -- * **Event** An __event__ can be any subset of the sample space `\(\Omega\)`. -- * **Event space** The set of all events in the sample space `\(\Omega\)` is called the __event space__, and is denoted `\(\mathcal{F}\)` -- * **Power set** Let `\(\Omega\)` be a set. The set of all possible combinations of the elements in `\(\Omega\)` is called the __power set__, and is denoted `\(2^\Omega\)`. --- * **Discrete random variable** A __discrete random variable__ `\(X\)` has a finite number of distinct outcomes. For example, rolling a die is a random variable with 6 distinct outcomes. For `\(\Omega\)` the sample space of `\(X\)`, `\(\Omega\)` contains a countable number of elements. -- A __Bernoulli random variable__ is a random variable that takes values in `\(\{0,1\}\)`, with the probability `\(p\)` (parameter) of taking 1. -- * **Continuous random variable** A __continuous random variable__ can take a continum of values within some interval (infinitely many values). For example, rainfall in Melbourne in May can be any number in the range from 0.00 to 200.00 mm. For any `\(\omega \in \Omega\)`, `\(\Pr(\omega)=0\)` -- <img src="ETC3460_slides_S1_2018_files/figure-html/dj-1.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/dj-2.png" width="45%" height="45%" /> --- ###Discrete Random Variable Let `\(R(X)\)` denote the __range__ of the random variable `\(X\)`, i.e., the set of possible values that X can take. -- * **Probability mass function (pmf)** The __probability mass function__ for the random variable `\(X\)`, denoted `\(f(x)\)`, enumerates the probability `\(X=x\)` for all elements in `\(R(X)\)`. That is `$$f(x)=\Pr(x)\text{ and } f(x)=0 \text{ for all } x\not\in R(X)$$` -- * **Bernoulli random variable** A __Bernoulli random variable__ has range `\(R(X)={0,1}\)` and pmf `$$f(x)=p^x(1-p)^{1-x},\ \ \ \ p\in [0,1]$$` where `\(p\)` denotes the probability of success. We refer to `\(p\)` as the parameter of the Bernoulli random Variable. --- * **Estimation for Bernoulli** Mathematically, for `\(n\)` denoting the total nomber of observations, we can estimate `\(p\)` by `$$\hat{p}=\frac{\# \{r_t>0\}}{n}$$` <img src="ETC3460_slides_S1_2018_files/figure-html/djbernoulli-1.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/djbernoulli-2.png" width="45%" height="45%" /> --- * **Poisson Random Variable** A __Poisson random variable__ has range `\(R(X)=\{1,2,3,\ldots\}\)`. The pmf for the Possiob random variable is given by `$$f(x)=\frac{\lambda^xe^{-\lambda}}{x!}$$` where the parameter `\(\lambda\)` is referred to as the intensity parameter. `\(\lambda\)` governs the size of counts that are most likely to occur. Larger the `\(\lambda\)`, bigger the probability of getting large values. -- Given `\(\lambda=10\)`, calculate the probability of event `$$\Pr(X=11) = \frac{\lambda^xe^{-\lambda}}{x!} = \frac{10^{11}e^{-10}}{11!} = 0.1137364$$` <img src="ETC3460_slides_S1_2018_files/figure-html/poisson-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> --- ###Continuous Random Variable Continous random variables are governed by their __probability density function (pdf)__. * **Probability density function** A random variable `\(X\)` is called continuous if its range is un-countable infinite and there exists a non-negative-valued function `\(f(x)\)` defined on `\(\mathbb{R}\)` such that for any event `\(B\subset R(X)\)`, we have `$$\Pr(B)=\int_B f(x)dx\ge 0, f(x)=0 \text{ for all } x\not\in R(X)\\ \int_\Omega f(x)dx=1$$` --- * **Normal distribution** A random variable `\(X\)` has pdf `$$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$` `\(\mu\)` is the location parameter; `\(\sigma\)` is the scale parameter. Often, denoted as `\(X\sim N(\mu, \sigma^2)\)`. Special case: stansard normal distribution, `\(\mu=0\)` and `\(\sigma = 1\)` -- <img src="ETC3460_slides_S1_2018_files/figure-html/norm-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> --- ###Moments and Expectations * **Expectation** If `\(X\)` is a discrete random variable with pmf `\(f(x)\)`, then the expected value of `\(X\)`, denoted `\(\mathbb{E}[X]\)`, is given by `$$\mathbb{E}(X) = \sum_{x\in R(X)} xf(x)$$` -- If `\(X\)` is a continous random variable with pdf `\(f(x)\)`, then the expected value of `\(X\)`, denoted `\(\mathbb{E}[X]\)`, is given by `$$\mathbb{E}(X) = \int_{R(X)} xf(x)dx$$` -- `$$X\sim N(0,1)\Rightarrow \mathbb{E}[X]=0\\ X\sim N(\mu,\sigma^2)\Rightarrow \mathbb{E}[X]=\mu$$` `$$X\sim \mathcal{B}(p)\Rightarrow \mathbb{E}[X]=p$$` `$$X\sim \mathcal{P}(\lambda)\Rightarrow \mathbb{E}[X]=\lambda$$` --- Features we want to know 1. The expected return 2. The risk 3. The likelihood of returns being above or below the mean 4. The likelihood of returns -- * **Moments** For each integer `\(k\)`, the __$k$-th moment__ of `\(X\)` is `$$\mu_k=\mathbb{E}[X^k]$$` The `\(k\)`-th __central moment__ of `\(X\)` is `$$\bar\mu_k=\mathbb{E}[(X-\mu_1)^k]$$` The `\(k\)`-th __standardized moment__ of `\(X\)` is `$$\bar\mu_k^s=\mathbb{E}\left[\left(\frac{X-\mu_1}{\sqrt{\bar\mu_2}}\right)^k\right]$$` --- `$$X\sim N(0,1)\Rightarrow \mathbb{E}[X]=0 \text{ and }\mathbb{E}[X^2]=1\\ X\sim N(\mu,\sigma^2)\Rightarrow \mathbb{E}[X]=\mu\text{ and }\mathbb{E}[X^2]=\sigma^2+\mu^2$$` `$$X\sim \mathcal{B}(p)\Rightarrow \mathbb{E}[X]=p\text{ and }\mathbb{E}[X^2]=p(1-p)+p^2$$` `$$X\sim \mathcal{P}(\lambda)\Rightarrow \mathbb{E}[X]=\lambda\text{ and }\mathbb{E}[X^2]=\lambda+\lambda^2$$` --- Features we want to know 1. The expected return - `\(\mathbb{E}[r_t]\)` 2. The risk - `\(\bar\mu_2=\mathbb{E}[(r_t-\mu_1)^2]\)` 3. The likelihood of returns being above or below the mean - __skewness__ 4. The likelihood of returns - __kurtosis__ -- * **Skewness** Likelihood of extremes above or belew mean. `$$\bar\mu_3=\mathbb{E}[(r_t-\mu_1)^3]$$` `$$\bar\mu_3^s=\mathbb{E}\left[\left(\frac{r_t-\mu_1}{\sqrt{\bar\mu_2}}\right)^3\right]$$` -- For stock returns, negative skewness is more likely than positive skewness. --- * **Kurtosis** Likelihood of extremes `$$\bar\mu_4^s=\mathbb{E}\left[\left(\frac{r_t-\mu_1}{\sqrt{\bar\mu_2}}\right)^4\right]$$` -- Kurtosis of the standard normal random variable is exactly 3. "excess kurtosis": `\(\text{Ex.Kurt} = \bar\mu_4^s -3\)` - `\(\text{Ex.Kurt}<0\)`, thinner tails than normal - `\(\text{Ex.Kurt}>0\)`, thicker tails than normal --- * student-t distribution `$$\bar\mu_1^s=0\\ \bar\mu_2^s=1\\ \bar\mu_3^s=0$$` same as normal `$$\text{Ex.Kurt}=\bar\mu_4^s-3 = \frac{6}{v-4}>0$$` where `\(v\)` is the degree of freedom parameter --- * **Sample estimation** `$$\bar{x}=\sum^T_{t=1}\frac{x_t}{T}$$` `$$s^2=\frac{1}{T-1}\sum^T_{t=1}(x_t-\bar{x})^2 \\ s=\sqrt{s^2}$$` `$$SK=\frac{1}{T}\sum^T_{t=1}\left(\frac{r_t-\bar{r}}{s}\right)^3$$` `$$KT=\frac{1}{T}\sum^T_{t=1}\left(\frac{r_t-\bar{r}}{s}\right)^4\\ \text{E-KT} = KT - 3$$` --- ##Distribution ###Joint Distribution Multivariate random variables allow relationship between two or more random quantities to be modeled and studied. `$$\mathbf{X}=\left(\begin{array}{c} X_1 \\ X_2 \\ \vdots \\ X_n\end{array}\right)$$` --- A multicariate random variable `\(\mathbf{X}\)` is called continous if its range is countably infinite and there exists a non-negative-valued function `\(f(x_1,\ldots,x_n)\)` defined for all `\((x_1,\ldots,x_n)' \in \mathbb{R}^n\)` such that for any event `\(B\subset R(X)\)`, we have `$$\Pr(B)=\int\cdots\int_{(x_1,\ldots,x_n\in B)} f(x_1,\ldots,x_n)dx_1\ldots dx_n \ge 0 \\ f(x_1,\ldots,x_n)=0 \text{ for all }(x_1,\ldots,x_n)'\not\in R(X)\\ \Pr(\Omega)=\int\cdots\int_{(x_1,\ldots,x_n\in \Omega)} f(x_1,\ldots,x_n)dx_1\ldots dx_n =1$$` The function `\(f(.)\)` is called the __multivariate probability density function (pdf)__. --- Let `\(X_1\)` and `\(X_2\)` be individually standard normal. Than `\(\mathbf{X}=(X_1,X_2)'\)` is __bivariate standard normal__, with correlation `\(\rho\)`, if and only if `$$f(x_1, x_2)=\frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{(x_1+x_2)^2-2\rho x_1x_2}{2\sqrt{1-\rho^2}}}$$` -- Let `\(X_1\sim N(\mu_1,\sigma_1^2)\)` and `\(X_2\sim N(\mu_2,\sigma_2^2)\)` be individually normal. Then `\(\mathbf{X}=(X_1,X_2)'\)` is __bivariate normal__, with correlation `\(\rho\)`, if and only if `$$f(x_1, x_2)=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{-\frac{1}{2\sqrt{1-\rho^2}}\left[\frac{(x_1-\mu_1)^2}{\sigma_1^2}+\frac{(x_2-\mu_2)^2}{\sigma_2^2}-\frac{2\rho(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2}\right]}$$` -- If `\(\mathbf{X}=(X_1,X_2)'\)` is bivariate normal, then both `\(X_1\)` and `\(X_2\)` must be normal. --- ###Marginal Distribution If we have joint distribution of `\(\mathbf{X}\)`, we can dudece the distribution of any subset of `\(\mathbf{X}\)`. The marginal disteibution requires _integrating out the variables which are not of interest_. -- Let `\(\mathbf{X}=(X_1,X_2)'\)` has joint pdf `\(f(x_1, x_2)\)`. The __marginal pdf__ of `\(X_1\)` is given by `$$f(x_1)=\int^\infty_{-\infty}f(x_1,x_2)dx_2$$` The __marginal pdf__ of `\(X_2\)` is given by `$$f(x_2)=\int^\infty_{-\infty}f(x_1,x_2)dx_1$$` --- Let `\(\mathbf{X}=(X_1,X_2)'\)` be bivariate normal, with correlation `\(\rho\)`, with `$$f(x_1, x_2)=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{-\frac{1}{2\sqrt{1-\rho^2}}\left[\frac{(x_1-\mu_1)^2}{\sigma_1^2}+\frac{(x_2-\mu_2)^2}{\sigma_2^2}-\frac{2\rho(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2}\right]}$$` Then the marginal distributions for `\(X_1\)` and `\(X_2\)` are `\(X_1\sim N(\mu_1,\sigma_1^2)\)` and `\(X_2\sim N(\mu_2,\sigma_2^2)\)`. -- * **Independence of Joint RVs** If event `\(A\)` and `\(B\)` are independent `$$\Pr(A\cap B)=\Pr(A)\Pr(B)$$` Random variable `\(X_1\)` and `\(X_2\)` are independent if and only if `$$f(x_1, x_2)=f_1(x_1)f_2(x_2)$$` --- ###Condition Distributions Let `\(R_m = \mathbb{E}[r_t]\)` denote the expected return on a stock and let `\(R_f = \mathbb{E}[B_t]\)` denote the expected return on a risk-free asset Bt, say a short-term treasury bond. All of modern finance is interested in explaining the behavior of `\(R_m − R_f\)`, the __excess returns__. We may want to model the excess returns conditional on `\(R_f\)`. -- Let `\(\mathbf{X}=(X_1,X_2)'\)` be bivariate normal, with correlation `\(\rho\)`, with `\(X_1\sim N(\mu_1,\sigma_1^2)\)` and `\(X_2\sim N(\mu_2,\sigma_2^2)\)`. The __conditional pdf__ of `\(X_1\)` given `\(X_2\in B\)`, with `$$\Pr(X_2\in B)=\int_B f_2(x_2)dx_2>0$$` is `$$f(x_1|X_2\in B)=\frac{\int_B f(x_1, x_2)dx_2}{\int_B\int^\infty_{-\infty}f(x_1, x_2)dx_1dx_2} = \frac{\int_B f(x_1, x_2)dx_2}{\int_Bf_2(x_2)dx_2}$$` --- If `\(B\)` is a single point, for `\(B=\{b\}\)` and `\(f_2(b)>0\)`, `$$f(x_1|X_2=b)=\frac{f(x_1,b)}{f_2(b)}$$` -- If `\(X_1\)` and `\(X_2\)` are independent, we have the conditional distribution if `\(X_1\)`, given `\(X_2=b\)` : `$$f(x_1|X_2=b)=\frac{f(x_1,b)}{f_2(b)}=\frac{f_1(x_1)f_2(b)}{f_2(b)}=f_1(x_1)$$` If variables are independent, the conditional distribution is the marginal distribution -- Let `\(\mathbf{X}=(X_1,X_2)'\)` be bivariate normal `$$\mathbf{X}\sim \mathcal{N}\left(\left(\begin{array}{C}\mu_1\\ \mu_2\end{array}\right), \left[\begin{array}{cc}\sigma^2_{11} & \sigma_{12}\\ \sigma_{12} & \sigma^2_{22}\end{array}\right]\right)$$` Then the conditional distribution of `\(X_1\)` given `\(X_2=x_2\)` is also normally distribution. For `\(\rho=\frac{\sigma_{12}}{\sqrt{\sigma^2_{11}\sigma^2_{22}}}\)`, `$$f(x_1|X_2=x_2) = \mathcal{N}\left(\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}(x_2-\mu_2),\sigma^2_{11}(1-\rho^2)\right)$$` --- ###Conditional Expections Instead of using the marginal density in the standard expections, we use the __conditional density__. -- Let `\(\mathbf{X}=(X_1,X_2)'\)` with pdf `\(f(x_1, x_2)\)`. Let `\(\mathrm{g}(X_1)\)` be some function of `\(X_1\)`. Then, for marginal density `\(f_2(x_2) >0\)`, the conditional expectation of `\(\mathrm{g}(X_1)\)` given `\(X_2=x_2\)` is `$$\begin{aligned} \mathbb{E}[\mathrm{g}(X_1)|X_2=x_2]&=\int^\infty_{-\infty}\mathrm{g}(x_1)f(x_1|X_2=x_2)dx_1 \\ &=\int^\infty_{-\infty}\mathrm{g}(x_1)\frac{f(x_1,x_2)}{f_2(x_2)}dx_1\end{aligned}$$` a function of `\(X_2\)` --- * **Simple linear regression** `$$\mathbf{Y}=\beta_0+\beta_1\mathbf{Z}+\mathbf{\epsilon}$$` `$$\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{z}$$` `$$\hat{\beta}_1=\frac{\sum^n_{i=1}(z_i-\bar{z})(y_i-\bar{y})}{\sum^n_{i=1}(x_i-\bar{z})^2}=\frac{\hat\sigma_{12}}{\hat\sigma^2_z}$$` -- `$$\begin{aligned}\hat{y}_i &= \hat\beta_0+\hat\beta_1z_i \\ &= \bar{y}+\hat\beta_1(z_i-\bar{z})\\ &= \hat\mu_y +\frac{\hat\sigma_{12}}{\hat\sigma_z}(z_i-\hat\mu_x)\\ &=\hat\mu_y+\hat\rho\frac{\hat\sigma_y}{\hat\sigma_z}(z_i-\hat\mu_x), \ \ \hat\rho=\frac{\hat\sigma_{yz}}{\sqrt{\hat\sigma^2_y\hat\sigma^2_z}} \end{aligned}$$` -- A simple linear regression model is the same as if we had just assumed that `\(Y\)` and `\(Z\)` were bivariate normal. --- * **Law of Iterated Expectations (L.I.E.)** The order of taking expectations does not matter `$$\mathbb{E}[\mathrm{g}(X_1)]=\mathbb{E}[\mathbb{E}[\mathrm{g}(X_1)|X_2=x_2]]$$` -- Let `\(\mathbf{X}=(X_1,X_2)'\)` be bivariate normal `$$\mathbf{X}\sim \mathcal{N}\left(\left(\begin{array}{C}\mu_1\\ \mu_2\end{array}\right), \left[\begin{array}{cc}\sigma^2_{11} & \sigma_{12}\\ \sigma_{12} & \sigma^2_{22}\end{array}\right]\right)$$` `$$f(x_1|X_2) = \mathcal{N}\left(\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}(X_2-\mu_2),\sigma^2_{11}(1-\rho^2)\right)$$` `$$\begin{aligned}\mathbb{E}[X_1]&=\mathbb{E}[\mathbb{E}[X_1|X_2]]\\ &= \mathbb{E} \left[\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}(X_2-\mu_2) \right]\\ &=\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}\mathbb{E}(X_2-\mu_2)\\ &=\mu_1 \end{aligned}$$` --- `$$\begin{aligned}\mathbb{E}[X_1X_2] &=\mathbb{E}[\mathbb{E}[X_1X_2|X_2]]\\ &= \mathbb{E}[X_2\mathbb{E}[X_1|X_2]]\\ &=\mathbb{E}\left[X_2\left(\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}(X_2-\mu_2)\right)\right]\\ &=\mu_1\mu_2+\rho\frac{\sigma_{11}}{\sigma_{22}}\mathbb{E}[(X_2-\mu_2)^2]\\ &=\mu_1\mu_2 + \sigma_{12} \end{aligned}$$` `$$\begin{aligned}Cov(X_1,X_2) &=\mathbb{E}[X_1X_2]-\mu_1\mu_2\\ &= \sigma_{12} \end{aligned}$$` -- * **Random walk** Let `\(\epsilon_t, t\ge1\)` denote a time series of i.i.d. random variables with mean 0 and variance 1. A common model for the price of a stock is the "random walk". `$$P_t=P_{t-1}+\epsilon_t$$` Let$P_0=0$ and `\(\mathbb{E}[\epsilon_t]=0\)`, then `$$\mathbb{e}[P_t|P_{t-1}]=\mathbb{E}[P_{t-1}|P_{t-1}] +\mathbb{E}[\epsilon_t|P_{t-1}]=P_{t-1}$$` --- * **Conditional variance** The variance of random variable `\(Y\)`, conditional on `\(X\)` is given by `$$Var(Y|X)=\mathbb{E}[(Y-\mathbb{E}[Y|X])^2|X]$$` * **Law of Total Variance** `$$Var(Y)=\mathbb{E}[Var(Y|X)]+Var(\mathbb{E}[Y|X])$$` -- Let `\(\mathbf{X}=(X_1,X_2)'\)` be bivariate normal `$$\mathbf{X}\sim \mathcal{N}\left(\left(\begin{array}{C}\mu_1\\ \mu_2\end{array}\right), \left[\begin{array}{cc}\sigma^2_{11} & \sigma_{12}\\ \sigma_{12} & \sigma^2_{22}\end{array}\right]\right)$$` `$$f(x_1|X_2) = \mathcal{N}\left(\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}(X_2-\mu_2),\sigma^2_{11}(1-\rho^2)\right)$$` `$$\begin{aligned}\mathbb{V}[X_1] &= \mathbb{E}[\mathbb{V}[X_1|X_2]]+\mathbb{V}[\mathbb{E}[X_1|X_2]]\\ &=\sigma^2_{11}(1-\rho^2)+\mathbb{V}\left[\mu_1+\rho\frac{\sigma_{11}}{\sigma_{22}}(X_2-\mu_2) \right]\\ &=\sigma^2_{11}(1-\rho^2)+\rho^2\sigma^2_{11}\\ &=\sigma^2_{11} \end{aligned}$$` --- * **Conditional Moments** For each integer `\(k\)`, the `\(k\)`-th conditional moment of `\(Y\)` given `\(X\)`, is `$$\mathbb{E}[Y^k|X]$$` --- #Part 1 Asset Pricing ##Returns **Simple return** : from time `\(t-1\)` to `\(t\)` Simple gross return `$$1+ R_t =\frac{P_t}{P_{t-1}}$$` Simple net return `$$R_t =\frac{P_t}{P_{t-1}}-1=\frac{P_t-P_{t-1}}{P_{t-1}}$$` --- **Log return** : The natural logarithm of the simple gross return is called the log return `$$r_t=\ln{(1+R_t)}=\ln{\frac{P_t}{P_{t-1}}}=\ln{P_t}-\ln{P_{t-1}}$$` For small `\(R_t\)` , `\(r_t=\ln{(1+R_t)} \approx R_t\)` -- Log returns are approximately equal to net returns because if x is small, then `\(\log(1+x)\approx x\)`. The `\(k\)`-period log return is `$$\begin{aligned}r_t(k) &= \log(1+R_t(k))\\ &=\log((1+R_t)\cdots(1+R_{t-k+1}))\\ &= \log(1+R_t)+\cdots+\log(1+R_{t-k-1})\\ &=r_t+\cdots+r_{r-k-1} \end{aligned}$$` --- * **Risk** Risk is the chance that the return on an asset will differ from its expected return `\(\mathbb{E}[r_t]\)`. <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-1-1.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-1-2.png" width="45%" height="45%" /> --- * **Standardization** `$$r\sim \mathcal{N}(\mu, \sigma^2)$$` `$$\Pr(r<0)=\Pr\left(\frac{r-\mu}{\sigma}<-\frac\mu\sigma \right) = \Pr\left(z<-\frac\mu\sigma\right)$$` ``` say mean 0.10, variance 0.16? ``` --- * **Risk Aversion** When exposed to uncertaninty, __risk aversion__ is the behavior of individuals or investors attempting to lower uncertainty. -- This implies that in order for you to hold a more risky asset, you must be compensated with the possibility of a higher return. * **Risk and Return** The __Risk-Return trade-off (RRT)__ states that, to bear higher rish, an investor must be compensated with the _possibility_ of a higher return. This is often succinctly stated as: there is a positive relationship between risk and expected return. -- If information on past returns was informative about the behavior of future returns, this could be used to help us mitigate the risk of investing in certain assets. --- ##Classical Models of Returns * **The Normal Model of Returns** `$$R_t\sim i.i.d.\mathcal{N}(\mu,\sigma^2)$$` -- Problems: - Normal random variables can take any value - Losses are generally bounded - Stock prices can only be so large and can never be negative --- * **Log-Normal Returns** `$$r_t=\log(1+R_t)\sim i.i.d.\mathcal{N}(\mu, \sigma^2)$$` It follows that the simple gross returns `\((1+R_t)\)` is log normally distributed. -- Lognormal returns admit a general formula for the `\(k\)`-th period returns `\(\log(1+R_t(k))\)` is the sum of `\(k\)` independent normal random variables `\(\mathcal{N}(\mu, \sigma^2)\)` `$$\log(1+R_t(k))\sim \mathcal{N}(k\cdot \mu, k\cdot \sigma^2)$$` `$$\Pr(\log(1+R_t(k))<x) = \Phi\left(\frac{\log(x)-k\mu}{\sqrt{k\sigma^2}} \right)$$` --- * **Testing for Normality of log returns** 1. Compareing moments 2. qqplot <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-2-1.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-2-2.png" width="45%" height="45%" /> --- * **Random Walk Model** The mean and variance of a __random walk__, conditional on `\(P_0\)`, are `$$\mathbb{E}[P_t|P_0]=P_0 +\mu\cdot t$$` `$$\mathbb{V}[P_t|P_0] = \sigma^2\cdot t$$` -- `\(\mu\)` is call the drift and determines the general direction of the random walk. `\(\sigma\)` is the volatility and determines how much the random walk fluctuates about the mean `\(\mathbb{E}[P_0] + \mu \cdot t\)` Most price follow random walks: not predictiable but follow a (short term) trend. --- * **Geometric Random Walks** `$$\begin{aligned}\log(1+R_t(k))&=r_1+\cdots+r_{t-k-1}\\ \frac{P_t}{P_{t-k}} = 1+R_t(k) &= e^{r_1+\cdots+r_{t-k-1}} \end{aligned}$$` -- taking `\(k=t\)` yields `$$P_t=P_0e^{r_1+\cdots+r_{t-k-1}}$$` -- __Geometric random walk__: log returns are i.i.d. normal with mean `\(\mu\)` and variance `\(\sigma^2\)` (parameters) The price process `\(\{P_t:t=1,2,\ldots\}\)` is said to follow the exponential of a random walk. -- The geometric random walk model implies that future price changes are independent of the past and therefore not possible to predict. - Positive trend - Future deviations from this upward trend cannot be predicted --- ##Portfolios Random walk indicates at least on an individual level, predicting the behavior of returns is very difficult. -- * Diversification in econometrics - Averaging `$$E(\frac{1}{2}(X_1+X_2))=\frac{1}{2}(\mu +\mu) = \mu$$` -- `$$\begin{aligned} Var(\frac{1}{2}(X_1+X_2))&=\frac{1}{4}Var(X_1) + \frac{1}{4}Var(X_2)\\ &=\frac{1}{4}(\sigma^2 +\sigma^2) = \sigma^2 /2 \end{aligned}$$` -- * Sampling and Distribution Refer to [here](https://fya.netlify.com/mean_distribution_converge.pdf) --- * **Portfolios** A __portfolio__ is simply a collection of assets, such as stocks, bonds, etc. Portifolios allow investors to mitigate risk, to some extent. -- Goals 1. maximize the expected return 2. minimize risk Choosing the allocation of assets, weights, so that we simultaneously maximize expected returns, and minimize risk. -- Risk measure: the standard deviation of the return on our portfolios. -- ###Combining one Risky and one Riskless Asset Return on a risky asset: `\(R\)` Expected value: `\(\mu_R\)` Return on a riskless asset (e.g. one-month U.S. treasury bill): `\(R_f\)` Expected value: `\(\mu_f\)` Finding a allocation rule (weight) that's optimal. `\(w \in [0,1]\)` --- * **Optimal Allocation** 1. Return on such a portfolio `$$R_p=wR+(1-w)R_f$$` 2. The expected return on the portfolio `$$E(R_p)=w\mu_R+(1-w)\mu_f$$` 3. The variance of the portfolio `$$V[R_p]=w^2\sigma^2_R+(1-w)^2\sigma^2_{R_f}$$` 4. Riskless asset with no risk `\(E[R_f]-\mu_f\)` `\(V(R_f)= 0\)` `$$V[R_p]=w^2\sigma^2_R$$` --- Determine `\(w\)` by decide either the expected return or risk one wishes to take. i.e. Determine `\(E(R_p)\)` or `\(V(R_p)\)` `$$w=\frac{\mu_{R_p}-\mu_f}{\mu_R-\mu_f}$$` `\(\mu_R-\mu_f\)` is referred to as the _excess return_ `$$w=\frac{\sigma_{R_p}}{\sigma_R}$$` -- `$$\begin{aligned}E(R_p)&=w\mu_R+(1-w)\mu_f\\ &=\frac{\sigma_{R_p}}{\sigma_R}\mu_R+(1-\frac{\sigma_{R_p}}{\sigma_R})\mu_f\\ &= \mu_f +\frac{\sigma_{R_p}}{\sigma_R}(\mu_R-\mu_f)\end{aligned}$$` --- * **Conditional Market Line** `$$\mu_{R_p} = E(R_p) = \mu_f +(\mu_R -\mu_f)\frac{\sigma_{R_p}}{\sigma_R}$$` When the risky asset R is the "market portfolio", the above equation is called __the capital market line__ The CML shows how `\(\mu_{R_p}\)` depends on `\(\sigma_{R_p}\)` -- Slope: `\(\frac{\mu_R-\mu_f}{\sigma_R}\)` `\(\mu_R-\mu_f\)` can be interpreted as a "risk-premium" on asset R The slope of the CML is the ratio of the risk-premium to the standard deviation of the market portfolio. -- For a 1-unit change in risk `\(\sigma_{R_p}\)`, changes in expected excess return: `\(E(R_p-\mu_f)-\mu_R-\mu_f\)` -- * **Sharp ratio** Slope: `\(\frac{\mu_R-\mu_f}{\sigma_R}\)` Measures the relative performance associated with investing. --- ### Two Risky Assets `$$R_p=wR_1+(1-w)R_2$$` `$$E(R_j)=\mu_j,\ \ \ V(R_j)=\sigma^2_j, \ \ \ \sigma_{12}=Cov(R_1,R_2), \ \ \ \text{for} j=1,2$$` Expected return `$$\mu_{R_p}=E(R_p)=w\mu_{R_1}+(1-w)\mu_{R_2}$$` The protfolios risk `$$\begin{aligned} \sigma^2_{R_p}&=w^2V(R_1)+(1-w)^2V(R_2)+2w(1-w)Cov(R_1, R_2)\\ &=w^2\sigma_1^2 +(1-w)^2\sigma_2^2+2w(1-w)\sigma_{12} \end{aligned}$$` -- Finding a `\(w\)` that minimizes risk `$$\underset{w\in[0,1]}{\min}\sigma^2_{R_p}$$` Solution (first-order condition) `$$\hat w=\frac{\sigma^2_2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}$$` --- `$$\begin{aligned} R_p&=\hat wR_1+(1-\hat w)R_2\\ &= R_2+(R_1-R_2)\hat w\\ &=R_2+(R_1-R_2)\frac{\sigma^2_2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}} \end{aligned}$$` `$$\begin{aligned}\mu_{R_p}&=\hat w\mu_1+(1-\hat w)\mu_2\\ &=\mu_2+(\mu_1-\mu_2)\hat w\\ &=\mu_2+(\mu_1-\mu_2)\frac{\sigma^2_2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}} \end{aligned}$$` --- `$$\mu_{R_p}=\mu_2+(\mu_1-\mu_2)\frac{\sigma^2_2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}$$` * **Conditional Optimal Portfolio Allocation** Define `$$X=R_2-R_1\ \ \ \ \ \ \ \ \ \ \ \ \ Y=R_2$$` `$$Cov(Y,X)=\sigma_2^2-\sigma_{12}$$` `$$V(X) = \sigma_1^2+\sigma_2^2-2\sigma_{12}$$` `$$\begin{aligned}\mu_{R_p}&=\mu_2+(\mu_1-\mu_2)\frac{\sigma^2_2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}\\ &=\mu_y+\mu_x\frac{Cov(Y,X)}{Var(X)}\\ &=\mu_y+\beta_1\mu_x \end{aligned}$$` --- `$$\mu_{R_p}=\mu_y+\beta_1\mu_x$$` * **Linear regression** `$$Y=\beta_0+\beta_1X+\epsilon$$` `$$\beta_0=\mu_y-\beta_1\mu_x=\mu_{R_p}=w\mu_1+(1-w)\mu_2$$` `$$\beta_1=\frac{Cov(X,Y)}{Var(X)}=w$$` -- Restate the minimum variance portfolio optimization `$$\begin{aligned}R_y&=\mu_p+wR_x+\epsilon\\ &=\beta_0+\beta_1R_x+\epsilon\\ R_2&=\beta_0+\beta_1(R_2-R_1)+\epsilon\end{aligned}$$` "Outcome": Variable `\(R_2\)` "Covariate": `\(R_2-R_1\)` --- * Example `$$\widehat{R_{IBM}}=\underset{(6e-04)}{0.0011}+\underset{(0.0353)}{0.3336}(R_{IBM} - R_{GOOG})$$` ``` ## ## Call: ## lm(formula = ibmr ~ I(ibmr - googr)) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.022239 -0.005330 -0.000987 0.003966 0.045570 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.0010807 0.0005805 1.862 0.0642 . ## I(ibmr - googr) 0.3336115 0.0352837 9.455 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.008183 on 197 degrees of freedom ## Multiple R-squared: 0.3121, Adjusted R-squared: 0.3087 ## F-statistic: 89.4 on 1 and 197 DF, p-value: < 2.2e-16 ``` --- ##Capital Asset Pricing Model The CAPM model procides estimates of expected rates of return ono individual investments by comparing them against "the market": what is the "fair" rate of return on invested capital. * **Notation** `\(r_{it}\)` return on an individual asset at time t `\(r_{ft}\)` return on a riskless asset at time t (e.g. short-term treasury bond) `\(r_{mt}\)` return on "the market" at time t, a weighted portfolio of all market activity (can be represented by e.g. Dow Jones Industrial Average, S&P500) -- * **Beta risk** `$$\begin{aligned}\beta&=\frac{E[(r_{it}-r_{ft}-(\mu_i-\mu_f))(r_{mt}-r_ft-(\mu_m-\mu_f))]}{V(r_{mt}-r_{ft})}\\ &= \frac{Cov(r_{it}-r_{ft}, r_{mt}-r_{ft})}{V(r_{mt}-r_{ft})} \end{aligned}$$` --- * **CAPM** The CAPM is a simple linear regression model `$$(r_{it}-r_{ft})=\alpha+\beta(r_{mt}-r_{ft})+\epsilon_t$$` * **Security Characteristic Line** Above regression line sometimes is called the __Security Characteristic Line__. This line characterizes the performance of a given asset against that of the market at every point in time. --- ###CAPM Interpretation * **BETA** We can classify individual stocks, or portfolios of stocks, according to their degree of beta risk `\(\beta\)` `$$\begin{array}{rl}\text{Aggressive}&\beta>1\\ \text{Tracks the market}& \beta=1\\ \text{Conservative}&0<\beta<1\\ \text{Independent of the market}&\beta=0\\ \text{Imperfect Hedge}&-1<\beta<0\\ \text{Perfect Hedge}& \beta=-1 \end{array}$$` -- * **ALPHA** Besides `\(\beta\)`-risk, the CAPM captures an additional source of risk called `\(\alpha\)`-risk. `\(\alpha\)`-risk refers to an assets ability to earn abnormal returns relative to the market return. `$$\begin{array}{rl}\text{Inadequate Reward for assumed risk}&\alpha<0\\ \text{Adequate Reward for assumed risk}& \alpha=1\\ \text{Excess Reward for assumed risk}&\alpha>0\end{array}$$` --- * **CAPM Example** `$$\widehat{(r_{IBM}-r_f)}=\underset{(7e-04)}{8e-04}+\underset{(0.1212)}{0.1774}(r_{m} - r_f)$$` ``` ## ## Call: ## lm(formula = I(ibmr - rf) ~ I(djr - rf)) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.029657 -0.006416 -0.000731 0.005872 0.033101 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.0007901 0.0006960 1.135 0.258 ## I(djr - rf) 0.1774286 0.1212176 1.464 0.145 ## ## Residual standard error: 0.009813 on 197 degrees of freedom ## Multiple R-squared: 0.01076, Adjusted R-squared: 0.005737 ## F-statistic: 2.142 on 1 and 197 DF, p-value: 0.1449 ``` --- * **Risk** CAPM decomposes risk into teo components: _systematic risk_ and _idiosyncratic risk_. `$$\begin{aligned}E[(r_{it}-r_{ft})^2]&=E[(\alpha+\beta(r_{mt}-r_{ft}))^2] +E[\epsilon^2_t]+\underset{=0}{\underbrace{2Cov(\epsilon_t, (\alpha+\beta(r_{mt}-r_{ft})))}}\\ &=\underset{\text{Systematic Risk}}{\underbrace{E[(\alpha+\beta(r_{mt}-r_{ft}))^2]}} +\underset{\text{Idiosyncratic Risk}}{\underbrace{E[\epsilon^2_t]}} \end{aligned}$$` -- Systematic risk is also known as non-diversifiable risk. Idiosyncratic risk represents the diversifiable risk. Standard error of the regression `\(\hat\sigma_\epsilon\)` provides an estimate of the idiosyncratic risk of the asset. Model `\(R^2\)` provides an estimate of the total risk that is due to systematic risk non-diversifiable, and `\(1-R^2\)` represents the proportion of idiosyncratic risk, which can not be diversified away. --- ###Fama-French 3 Factor Model Including two additional risk factors to explain investment returns (size and value). * **Size** __Size__, or __SMB__ (small minus big) is the difference between the return on a portfolio of small stocks (in terms of _market capitalization_) and the return on a portfolio of big stocks (the performance of big stocks versus the of small stocks). -- __Market capitalization__ is the market value at a point in time of the shares outstanding. Market capitalization is equal to the share price times the number of shares outstanding. -- Incorporating SMB into the CAPM shows whether management was relying the small firm effect (investing in stocks with low market capitalization) to earn an abnormal return. --- * **Value** __Value__, or __HML__ (high minus low) is the difference between the return on a portfolio of high book-to-market stocks and the return on a portfolio on a portfolio of small book-to-market stocks (the performance of "value" stocks relative to growth stocks) -- __Book-to-market__ ratio is defined as `$$\text{B-to-M}=\frac{\text{book value of firm}}{\text{market value of firm}}$$` Book value is calculated by looking at the firm's historical cost, or accounting value. Market value is determined in the stock market through its market capitalization. --- * **Fema-French 3 Factor Model** `$$r_{it}-r_{ft}=\alpha+\beta_1(r_{mt}-r_{ft})+\beta_2SMB_t+\beta_3HML_t+\epsilon_t$$` -- __Interpretation__ for `\(\beta_2\)`: an estimated value greater than 0.5 signifies a portfolio composed mainly of small cap stocks, and a zero value signifies large cap stocks. __Interpretation__ for `\(\beta_3\)`: an estimated value greater than 0.3 signifies a portfolio composed mainly of value stocks. -- Example `$$\widehat{r_{it}-r_{ft}}=0.37+1.22(r_{mt}-r_{ft})+0.10SMB_t+0.73HML_t$$` -- * **Multi-Factor CAPM** Addtional factor called "Momentum" * **Momentum** __Momentum__ captures returns constructed bu buying stocks with high returns and selling stocks with low returns over the same period. This factor captures hedging behavior of investors. -- `$$r_{it}-r_{ft}=\alpha+\beta_1(r_{mt}-r_{ft})+\beta_2SMB_t+\beta_3HML_t+\beta_4MOM_t+\epsilon_t$$` Can test significance. --- ###Properties of CAPM regression model If the properties of the model are not satisfied, then the model will not be correct, inference will be misleading. `$$\begin{aligned}E[\epsilon_t]&=0\\ E[\epsilon^2_t]&=\sigma^2_\epsilon\\ E[\epsilon_t\epsilon_{t-j}]&=0\forall j\neq i\\ E[\epsilon_t(r_{mt}-r_{ft})]&=0 \end{aligned}$$` --- `\(E[\epsilon_t]=0\)`: the idiosyncratic risk has zero mean. -- `\(E[\epsilon^2_t]=\sigma^2_\epsilon\)`: the variance is constant across time. - homoskedasticity - may hold for low frequency data -- `\(E[\epsilon_t\epsilon_{t-j}] =0\forall j\neq i\)`: uncorrelated through time -- `\(E[\epsilon_t(r_{mt}-r_{ft})] =0\)`: exogeneity - all systematic risk is explained by the market factor - required to decompose total risk -- >Condiser: 1. Model fit 2. Diagnostics for the independent variable 3. Diagnostics for the disturbance term --- ###Model Fit If the model fits the data well, it should be the case that the model explain a significant portion of variation in the data. `$$R^2=\frac{\text{Explained sum of squares}}{\text{Total sum of squares}}=\frac{SSE}{SST}=\frac{\sum^{T}_{t=1}(\bar{y}-\hat{\alpha}-\hat{\beta}x_t)^2}{\sum^T_{t=1}(y_t-\bar y_t)^2}$$` -- For CAPM, `\(R^2\)` measures the proportion of total risk that is due to systematic risk. `\(1-R^2\)` measures idiosyncratic risk. --- * **Adjusted R squared** `$$\bar R^2=1-R^2\frac{T-1}{T-K-1}$$` `\(\bar R^2\)` penalizes the model for adding regressors which do not explain variability in the dependent variable. -- The standard CAPM can explain around 70% of return variabilty The three-factor CAPM may reach 90% --- ###Test for significance * **Single Parameter Tests** `$$\begin{aligned}H_0&:\beta_1=0[\text{Market factor is not priced}]\\ H_1&:\beta_1\neq0[\text{Market factor is priced}] \end{aligned}$$` -- `$$\begin{aligned}T&=\frac{\hat \beta-r}{s.e.(\hat\beta)}\\ s.e.(\hat\beta)&=\sqrt{\frac{\hat\sigma^2_\epsilon}{\hat\sigma_x^2}\frac{1}{T}}\\ \hat\sigma^2_\epsilon&=\frac{1}{T-1}\sum^T_{t=1}(x_t-\bar x_t)^2\\ \hat\sigma^2_\epsilon&=\frac{1}{T-1}\sum^T_{t-1}\hat\epsilon_t^2 \end{aligned}$$` -- `$$\begin{aligned}T&\overset{asy}\sim N\text{ under }H_0\\ T&\sim t_{T-K-1}\text{ under }H_0\end{aligned}$$` --- * **Joint Parameter Tests** `$$\begin{aligned}H_0&:\beta_2=\beta_3=\beta_4=0\\ H_1&:\text{at least one is non-zero} \end{aligned}$$` -- `$$J=\frac{RSS_{CAPM}-RSS_{\text{multi-}CAPM}}{RSS_{\text{multi-}CAPM}/(T-K-1)}\sim\chi^2_{l}$$` --- ###Diagnostics for the Disturbance Term `$$\begin{aligned}E[\epsilon_t]&=0\\ E[\epsilon^2_t]&=\sigma^2_\epsilon\\ E[\epsilon_t\epsilon_{t-j}]&=0\forall j\neq i\\ E[\epsilon_t(r_{mt}-r_{ft})]&=0 \end{aligned}$$` -- __Lagrange Multiplier (LM)__ tests are regression based tests. They come from a general framework and can be adapted to different situation. -- * **LM Tests** If the model is correct, there should not be structures in the error term. `$$\begin{aligned} E[\epsilon^2_t]=\sigma^2_\epsilon:\text{Homoskedasticity}\\ E[\epsilon_t\epsilon_{t-j}]=0\forall j\neq i:\text{No Autocorrelation}\\ E[\epsilon_t(r_{mt}-r_{ft})]=0:\text{Exogeneity} \end{aligned}$$` `\(E[\epsilon_t]=0\)` is satisfied by definitin of the CAPM. --- * **Homoskedasticity** `$$\begin{aligned}H_0&:\sigma_\epsilon^2\text{ is constant}[\text{Homoskedasticity}]\\ H_1&:\sigma_\epsilon^2\text{ is not constant}[\text{Heteroskedasticity}] \end{aligned}$$` If heteroskedasticity, standard errors are not correct: significant tests are not reliable, conclusion for `\(\beta\)` and `\(\alpha\)` incorrect. -- * **Whits's test** Auxiliary regression `$$\hat\epsilon^2_t=\gamma_0+\gamma_1x_t+\gamma_2x^2_t+v_t$$` `$$\begin{aligned}H_0&:\gamma_1=\gamma_2=0\\ H_1&:\text{at least one is non-zero} \end{aligned}$$` `$$W=T\cdot R^2 \overset{asy}{\sim}\chi^2_2$$` --- * **Testing for ARCH** __Autoregressive conditional heteroskedasiticity (ARCH)__: large volatility are followed by periods of large volatility (volatility clustering). -- Auxiliary regression `$$\hat\epsilon^2_t=\gamma_0+\gamma_1\hat\epsilon^2_{t-1}+\gamma_2\hat\epsilon^2_{t-2}+\cdots+\gamma_p\hat\epsilon^2_{t-p}+v_t$$` `$$\begin{aligned}H_0&:\gamma_1=\gamma_2=\cdots=\gamma_p=0[\text{No ARCH}]\\ H_1&:\text{at least one is non-zero[ARCH]} \end{aligned}$$` `$$ARCH(p)=R^2\cdot T\overset{asy}{\sim}\chi^2_p$$` -- ARCH implies there exists some persistent behavior in the volatility of returns that can be modelled. --- * **Test for Autocorrelation** If autocorrelation in error, standard errors are not correct: significant tests are not reliable, conclusion for `\(\beta\)` and `\(\alpha\)` incorrect. Auxiliary regression `$$\begin{aligned}\hat\epsilon_t=&\gamma_0+\gamma_1x_{1t}+\cdots+\gamma_kx^2_{kt}\\ &+\rho_1\hat\epsilon_{t-1}+\rho_2\hat\epsilon_{t-2}+\cdots+\rho_p\hat\epsilon_{t-p}+v_t\end{aligned}$$` `$$\begin{aligned}H_0&:\rho_1=\rho_2=\cdots=\rho_p=0\\ H_1&:\text{at least one is non-zero} \end{aligned}$$` `$$AR(p)=T\cdot R^2 \overset{asy}{\sim}\chi^2_p$$` --- * **HAC error** `$$Var(\mathbf{\hat{\beta}}|\mathbf{X})=(\mathbf{X'X})^{-1}[\mathbf{X}'Var(\mathbf{u}|\mathbf{x})\mathbf{X}](\mathbf{X'X})^{-1}$$` With Homo `$$Var(\mathbf{u}|\mathbf{X})=\sigma^2 \mathbf{I_n} \Rightarrow Var(\mathbf{\hat{\beta}}|\mathbf{X})=\sigma^2(\mathbf{X'X})^{-1}$$` -- With HTSK `$$Var(\mathbf{\hat{\beta}}|\mathbf{X})=(\mathbf{X'X})^{-1}\left[\mathbf{X}'\left( \begin{array}{cccc} \sigma^2_1 &0 & \cdots & 0 \\ 0 & \sigma^2_2 & \cdots & 0\\ \vdots &\vdots&\ &\vdots\\ 0 & 0 & \cdots & \sigma_n^2\end{array}\right) \mathbf{X}\right] (\mathbf{X'X})^{-1}$$` -- With HTSK and serial correlation `$$Var(\mathbf{\hat{\beta}}|\mathbf{X})=(\mathbf{X'X})^{-1}\left[\mathbf{X}'n\mathbf{\Lambda}_n \mathbf{X}\right] (\mathbf{X'X})^{-1}$$` `\(\mathbf{\Lambda}_n\)` is an estimator of the long run variance of `\(\mathbf{u}\)` --- * **Engodeneity** If `\(x_t\)` is endogenous, the OLS estimation coefficients are __biased and inconsistent__. -- `$$E[\epsilon_t(r_{mt}-r_{ft})]=0$$` implication: the error term in any time period t is uncorrelated with each of the regressors in all time periods, past, present, and future. `$$Corr(u_t,x_{11})=\cdots=Corr(u_t,x_{T1})=Corr(u_t,x_{1k})=\cdots=Corr(u_t,x_{Tk})=0$$` Only restrict correlation between error terms and regressors, but not regressors themselves or errors themselves. --- ##Efficient Market Hypothesis __EMH__ implies - The current price of an asset reflects all information available in the market -- - The current price provides no information regarding future asset movements -- - Future returns are completely unpredictable, given information on past returns -- - Traders can not systematically use newly arriving information to make a profit - Conditional on all previous information, returns are completely random -- In a word, you can't beat the market. --- * **Notation** `\(y_t\)` the variable of interest (returns or prices or whatever) `\(\mathcal{F}_{t-1}\)` information available to date * **Example in CAPM** `$$y_t=R_{it}-R_{ft}$$` `\(\epsilon_t\)` represents idiosyncratic risk `$$E[y_t|\mathcal{F}_{t-1}]=\alpha+\beta(R_{it}-R_{ft})$$` --- * **Random Walk** `$$y_t=\mu+\epsilon_t$$` Example `\(y_t\)` log-returns - `\(\epsilon\sim N(0,1)\)` EMH satisfied - `\(\epsilon\sim t_v\)` EMH satisfied `\(y_t=P_t\)` - `\(E[y_t|\mathcal{F}_{t-1}]=P_t-1\)` --- * **White Noise** `\(e\sim WN(0, \sigma^2)\)` if > a) `\(E(e_t)=0\forall t\)` b) `\(Var(e_t)=\sigma \forall t\)` c) `\(Cov(e_t, e_{t-j})=0 \forall j\neq0\)` (no linear relationship) If it's also normally distributed -- Gaussian white noise <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-3-1.png" width="60%" height="60%" style="display: block; margin: auto;" /> --- If we assume the error term is white noise, we can test the EMH - return predictability - variance of asset returns over different time horizon -- If markets are efficient, them the variance of asset returns over different time horizons should roughly increase with their horizon. The variance of `\(n\)` period returns should simply be `\(n\)` times the variance of the 1-period return. --- ###Return Predictability Mesure return predictability through serial dependence. -- __Autocorrelation function (ACF)__: measure of sample serial dependence `$$\hat\gamma(k)=\frac{\sum^T_{t-k+1}(r_t-\bar r)(r_{t-k}-\bar r)}{\sum^T_{t=1}(r_t-\bar r)^2}$$` `$$\gamma(k)=\frac{Cov(r_t, r_{t-k})}{V(r_t)}$$` -- No correlation implies no predictability and EMH satisfied. --- * **Stationary Time Series** A univariate time series is an ordered squance of random variables indexed by time. (Infinite number of realizations) `\(\{y_t:t=\cdots -2, -1, 0, 1, 2, \cdots \}\)` **Weakly Stationary** (covariance stationary, second-order stationary) > a) `\(E(y_i) =\mu < \infty \ \ for\ all\ t\)` b) `\(Var(y_t)= E[(y_t-\mu)^2]=\gamma_0<\infty\ \ for\ t\)` c) `\(Cov(y_t)= E[(y_t-\mu)(y_{t-j}-\mu)]=\gamma_j < \infty\)` Its first and second moments are both finite and time invariant. The covariance depends only on the time interval separating them and not on time itself) -- <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-4-1.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-4-2.png" width="45%" height="45%" /> --- <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-5-1.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-5-2.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-5-3.png" width="45%" height="45%" /><img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-5-4.png" width="45%" height="45%" /> --- * **Single Test** If `\({y_t}\)` is a stationary, when the sample size `\(T\)` is large, `\(\hat\gamma(k)\)` should be approximately normal `\(\hat\gamma(k)\sim N(0,1/T)\)` and `\(\sqrt{T}\hat\gamma(k)\sim N(0,1)\)` `$$\begin{aligned}H_0&:\gamma(k)=0\\ H_1&:\gamma(k)\neq0 \end{aligned}$$` -- * **Test for more orders** Auxiliary regression `$$\hat{\epsilon}_t=\gamma_0 +\gamma_1\hat{\epsilon}_{t-1}+\cdots+\gamma_k\hat{\epsilon}_{t-k} +v_t$$` `$$\begin{aligned}H_0 &: \gamma_1=\gamma_2=\cdots=\gamma_k=0\\ H_1 &: \gamma_j \neq 0\ \text{for at least one }\ j=1, 2, 3, \cdots , k\end{aligned}$$` `$$AR(k)=T\cdot R^2 \overset{asy}\sim \chi_k^2 \text{ under } H_0$$` --- ###Variance Ratio To determnine predictability by comparing variance of asset returns over different time horizon `$$\begin{aligned}s^2_1&=\frac1T\sum^T_{t=1}(r_t-\bar r)^2\\ s^2_n&=\frac1T\sum^T_{t=1}(r_{n,t}-\bar r_n)^2\\ \end{aligned}$$` -- If no autocorrelation `$$VR_n=\frac{s^2_n}{n\cdot s^2_1}\approx1$$` -- `$$VR_n=\left\{\begin{array}{cc}=1 & [\text{No autorr.}]\\ >1&+\text{autorr.}\\ <1&-\text{autorr.} \end{array}\right.$$` The assumption that variance of returns are the same is part of stationarity. --- ###Autocorrelation Onle linear relationship -- `$$\hat\gamma^2(k)=\frac{\sum^T_{t-k+1}(r^2_t-\bar r^2)(r^2_{t-k}-\bar r^2)}{\sum^T_{t=1}(r^2_t-\bar r^2)^2}$$` measures correlation between the variance of returns at time `\(t\)`, and the variance of returns at time `\(t-k\)` -- While _mean_ returns exhibit little to no predictability, it is possible that _the variance of returns_ exhibit predictability. Not in the region of EMH. -- Apply to other moments and functions. --- ##Modeling Predictavle Returns When EMH is not satified, we can try to predict returns. We want to model the dynamics of returns and use them to generate forecast distribution ( _Conditional distributions_ ). -- ###AR processes * **AR(1) process** `$$y_t=c+\phi_1y_{t-1}+\epsilon_t$$` where `\(\epsilon_t\sim WN(0,\sigma^2_\epsilon)\)` and `\(V(\epsilon_t|\mathcal{F}_{t-1})=V(\epsilon_t)\)` -- An AR(1) process is __covariance (weakly) stationary__ if `\(|\phi_1|<1\)` --- * **Conditional Mean** `$$E(y_t|\mathcal{F}_{t-1})=E[c+\phi_1y_{t-1}+\epsilon|\mathcal{F}_{t-1}] = c+\phi_1y_{t-1}$$` * **One-step ahead point forecast** `$$\widehat{E}(y_{t+1}|\mathcal{F}_{t-1})=\hat c+\hat \phi_1 y_t$$` -- `$$(\hat c, \hat\phi_1)=\arg \min\sum^{T-1}_{t=1}(y_{t+1}-c-\phi_1y_t )^2$$` --- * **Conditional Variance** `$$\begin{aligned}V(y_t|\mathcal{F}_{t-1})&= V(c+\phi_1y_{t-1}+\epsilon_t|\mathcal{F}_{t-1})\\ &=V(c+\phi_1y_{t-1}|\mathcal{F}_{t-1})+V(\epsilon_t|\mathcal{F}_{t-1})\\ &=0 +V(\epsilon_t|\mathcal{F}_{t-1})\\ &=V(\epsilon_t)=\sigma^2_\epsilon \end{aligned}$$` -- * **One step ahead forecast variance** `$$\widehat{V}(y_{t+1}|\mathcal{F}_t)=\hat\sigma^2_\epsilon= \frac{\sum^{T-1}_{t=1}(y_{t+1}-\hat c- \hat\phi_1y_t)^2}{T-2}$$` This the estimated variance of the regression of `\(y_{t+1}\)` on a constant `\(c\)` and the regressor `\(y_t\)` --- * **Two-step ahead forecast** `$$\begin{aligned}E(y_{t+2}|\mathcal{F}_t)&=E[c+\phi_1y_{t+1}+\epsilon_{t+2}|\mathcal{F}_t]\\ &=c+\phi_1E[y_{t+1}|\mathcal{F}_t]+E[\epsilon_{t+2}|\mathcal{F}_t]\\ &=c+\phi_1(c+\phi_1y_{t-1})\\ &=c(1+\phi_1)+\phi^2_1y_t \end{aligned}$$` -- * **Three-step ahead forecast** `$$\begin{aligned}E(y_{t+3}|\mathcal{F}_t)&=E[c+\phi_1y_{t+2}+\epsilon_{t+3}|\mathcal{F}_t]\\ &=c+\phi_1E[y_{t+2}|\mathcal{F}_t]+E[\epsilon_{t+3}|\mathcal{F}_t]\\ &=c+\phi_1(c(1+\phi_1)+\phi^2_1y_t)\\ &=c(1+\phi_1+\phi_1^2)+\phi^3_1y_t \end{aligned}$$` etc... --- Taking __many steps ahead__ to the limit we have `$$\begin{aligned}\underset{h\rightarrow\infty}{\lim}E(y_{t+h}|\mathcal{F}_t)&=\underset{h\rightarrow\infty}{\lim}c(1+\phi_1+\cdots+\phi_1^h)+\underset{h\rightarrow\infty}{\lim}\phi_1^hy_t\\ &=\underset{\text{geometric series}}{\underbrace{c\sum^\infty_{h=0}\phi_1^h}}+y_t\left(\underset{h\rightarrow\infty}{\lim}\phi_1^h\right)\\ &=\frac{c}{(1-\phi_1)} \end{aligned}$$` Geometric series `$$a\sum^{n-1}_{k=0}r^k=a(\frac{1-r^n}{1-r})$$` -- For stationary models, this many steps ahead conditional expectation converges to the unconditional mean. --- * **Unconditional mean** `$$\begin{aligned}E(y_t)&=E(c+\phi_1y_{t-1}+\epsilon_t)\\ &=c+\phi_1E[y_{t-1}]+E[\epsilon_{t}]\\ &=c+\phi_1E[y_{t}]+E[\epsilon_{t}]\ \ (\text{Stationarity in }y_t) \\ \Rightarrow E(y_t)&=\frac c{1-\phi_1}=\underset{h\rightarrow\infty}{\lim}E(y_{t+h}|\mathcal{F}_t) \end{aligned}$$` -- * **Unconditional (long-run) Variance** `$$\begin{aligned}V(y_t)&=Var(c+\phi_1y_{t-1}+\epsilon_t)\\ &=\phi_1^2Var(y_{t-1})+\sigma^2_\epsilon\ \ (\text{Assuming }Cov(y_{t-1}, \epsilon_t)=0)\\ &=\phi_1^2Var(y_{t})+\sigma^2_\epsilon\ \ (\text{Stationarity in }y_t) \\ \Rightarrow Var(y_t)&=\frac{\sigma^2_\epsilon}{(1-\phi_1^2)}\neq \sigma^2_\epsilon=V(y_t|\mathcal{F}_{t-1}) \end{aligned}$$` -- `$$V(y_t)>V(y_t|\mathcal{F}_{t-1})$$` Use of information will lead to less uncertainty. --- * **Unconditional population autocorrelations** `$$Corr(y_t, y_{t-1})=\gamma(k)=\phi_1^k\text{ for }k=\ldots-1, 0, 1, 2, \ldots$$` `\(|\phi_1|\)` is a measure of 'persistence' -- * **AR(p) processes** A time series `\(y_t\)` for `\(t=\{\ldots-1, 0, 1, 2, \ldots\}\)` is an __AR(p) process__ if `$$y_t=c+\phi_1y_{t-1}+\phi_2y_{t-2}+\ldots+\phi_py_{t-p}+\epsilon_t$$` where `\(\epsilon_t\sim WN(0,\sigma^2_\epsilon)\)` and `\(V(\epsilon_t|\mathcal{F}_{t-1})=V(\epsilon_t)\)` --- For stationary AR(p) processes * **Unconditional mean** `$$E[y_t]=\frac{c}{(1-\phi_1-\ldots-\phi_p)}$$` * **Unconditional variance** `$$V(y_t)=\frac{\sigma^2_\epsilon}{\left(1-\sum^{i=p}_{i=1}\phi_i\rho_i\right)}$$` -- * For AR(1) `$$E(y_t)=\frac c{1-\phi_1}$$` `$$Var(y_t)=\frac{\sigma^2_\epsilon}{(1-\phi_1^2)}$$` --- * **Comments** Forecasts based on well-specified AR models are consistent (i.e. bias is negligible in large samples) Need to choose `\(p\)` - include enough lags to ensure that there is no left over serial correlation in the residuals. We can interpret `\(\epsilon_t\)` as 'news' (unpredictable market forces) --- ###Parametric Forecasting Strategy * **Returns** Decide AR(p) model for _returns_ (1) Estimated conditional mean `\(\hat E [r_{T+1}|\mathcal{F}_t]\)` (2) Estimated error variance `\(\hat V(\epsilon_{T+1})\)` -- (3) Assume a normal distribution for errors `\(\epsilon_{T+1}\sim N (0,\hat\sigma^2_\epsilon)\)` -- (4) Probabilities and Quantiles as needed `$$\begin{aligned}Pr(r_{T+1}<q|\mathcal{F}_t)&=Pr(r_{T+1}-\hat E [r_{T+1}|\mathcal{F}_t]<q-\hat E[r_{T+1}|\mathcal{F}_t]|\mathcal{F_t})\\ &=Pr(\epsilon_{T+1}<q-\hat E[r_{T+1}|\mathcal{F}_t]) \\ &=Pr(\frac{\epsilon_{T+1}}{\hat\sigma_\epsilon}<\frac{q-\hat E[r_{T+1}|\mathcal{F}_t]}{\hat\sigma_\epsilon})\\ &=Pr(z<\frac{q-\hat E[r_{T+1}|\mathcal{F}_t]}{\hat\sigma_\epsilon}) \end{aligned}$$` --- * **Prices** Given the __price__ `\(P_T\)` and assuming `\(r_{T+1}|\mathcal{F}_t\sim\mathbf{Normal}\)` Implies the conditional distribution for `\(P_{T+1}|\mathcal{F}_t\)` is __lognormal__ `$$\begin{aligned}Pr(P_{T+1}\leq q|\mathcal{F}_t) &= Pr(P_Te^{r_{T+1}}\leq q|\mathcal{F}_t )\\ &=Pr(r_{T+1}\leq \ln\left(\frac{q}{P_t}\right)|\mathcal{F}_t)\\ &=Pr\left(z\leq\frac{\ln\left(\frac{q}{P_t}\right)-\hat E[r_{T+1}|\mathcal{F}_t]}{\hat\sigma_\epsilon}\right) \end{aligned}$$` -- * **Test Normality - Jarque-Bera test** `$$\begin{aligned}H_0&:\text{Residuals are normal}\\ H_1&:\text{Residuals are not normal} \end{aligned}$$` `$$JB\sim \chi^2_2 \text{ under } H_0$$` --- ###Non-parametric Forecasting Strategy * **Returns** Decide AR(p) model for _returns_ (1) Estimated conditional mean `\(\hat E [r_{T+1}|\mathcal{F}_t]\)` (2) Estimated error variance `\(\hat V(\epsilon_{T+1})\)` -- (3) Use the empirical distribution (histogram) of the fitted residuals `\(\hat\epsilon_t=r_t- \hat e[r_t|\mathcal{F}_{t-1}],\text{ for }t=(p+1), (p+2),\ldots, T\)` (4) Probabilities and Quantiles as needed `$$\begin{aligned}\hat{Pr}(r_{T+1}\leq q |\mathcal{F}_t)&=\frac1{T-p}\sum^T_{t=p+1}1(\hat\epsilon_t \leq q-\hat{E}[r_{T+1}|\mathcal{F}_t]|\mathcal{F}_t)\\ &=\text{relative proportion of } \hat\epsilon_t \text{'s that satisfy inequality} \end{aligned}$$` --- #Part 2 Modeling Volatility When EMH is satisfied <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-6-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> Mean returns are much smaller than the standard deviation of returns. --- Our model for returns is `$$r_t=E[r_t|\mathcal{F}_{t-1}]+\epsilon_t$$` with `\(\epsilon_t\sim WN(0, \sigma^2_\epsilon)\)` > In AR(p) model > `$$E[r_t|\mathcal{F}_{t-1}]=c+\phi_1r_t+\phi_2r_{t-1}+\cdots+\phi_pr_{t-(p-1)}$$` > with `\(V(\epsilon_t|\mathcal{F}_{t-1})=\sigma^2_\epsilon\)` We will look at modeling the conditional variance as `$$V(\epsilon_t|\mathcal{F}_{t-1})= g(\mathcal{F}_{t-1})$$` --- ##Volatility __Volatility__ is a characterization of risk associated with an asset. (usually measured by the _standard deviation_) If volatility is high then asset prices are changing more repidly than when volatility is low. -- Interested in __conditional standard deviations__, associated with relatively high frequency returns. -- To compare volatility from returns over different frequencies (and at different times) `\(\Rightarrow\)` scale estimated volatility to an __annual__ frequency. __Annualized volatility__ is a scaled volatility measure obtained from higher frequency returns but scaled to reflect the period of one year. `$$\sqrt{252}\sigma_t$$` `\(\sigma_t\)` is the daily volatility. 252 trading days in a year. Ignore correlation. --- * **Testing for time varying volatility** - Plots of `\(\hat\epsilon_t^2\)` against time (time varying) - Autocorrelations for `\(\hat\epsilon^2_t\)` (depends on its past) -- `$$r_t=\mu_r+\epsilon_t$$` with `\(\epsilon_t\sim WN\)` If serial correlation in `\(\epsilon^2_t\)`, constant volatility models (e.g. CAPM) will be inadequate. If test shows serially correlation, some AR model maybe useful to model the conditional variance. --- * **An LM test for ARCH(1)** __Autoregressive conditional heteroskedasiticity (ARCH)__: large volatility are followed by periods of large volatility (volatility clustering). -- If the true model is `$$y_t=x_t^\prime\beta+\epsilon_t$$` with `$$\epsilon_t^2=\sigma^2+\rho_1\epsilon^2_{t-1}+u_t$$` but we estimate `$$y_t=x_t^\prime\beta+\epsilon_t$$` Then there will be a dynamic pattern remaining in the squared residuals. --- `$$\begin{aligned}H_0&:\rho_1=0\\ H_1&:\rho_1\neq0 \end{aligned}$$` 1. Regression `\(y_t=x_t^\prime\beta+\epsilon_t\)` `$$y_t=x_t^\prime\beta+\epsilon_t$$` -- 2. Auxiliary regression `$$\hat\epsilon^2_t=\gamma_0+\gamma_1\hat\epsilon^2_{t-1}+u_t$$` -- 3. Distribution `$$R^2\cdot T{\sim}\chi^2_1$$` --- * **An LM test for ARCH(q)** If the true model is `$$y_t=x_t^\prime\beta+\epsilon_t$$` with `$$\epsilon_t^2=\sigma_0^2+\rho_1\epsilon^2_{t-1}+\rho_2\epsilon^2_{t-2}+\ldots+\rho_q\epsilon^2_{t-q}+u_t$$` but we estimate `$$y_t=x_t^\prime\beta+\epsilon_t$$` Then there will be a dynamic pattern remaining in the squared residuals. --- `$$\begin{aligned}H_0&:\rho_1=\rho_2=\cdots=\rho_1=0\\ H_1&:\text{ at least one } \rho_k\neq0, \text{ for }k=1,2,\ldots,q \end{aligned}$$` 1. Regression `\(y_t=x_t^\prime\beta+\epsilon_t\)` `$$y_t=x_t^\prime\beta+\epsilon_t$$` -- 2. Auxiliary regression `$$\hat\epsilon^2_t=\gamma_0+\gamma_1\hat\epsilon^2_{t-1}+\gamma_2\hat\epsilon^2_{t-2}+\cdots+\gamma_q\hat\epsilon^2_{t-q}+u_t$$` -- 3. Distribution `$$R^2\cdot T{\sim}\chi^2_q$$` -- A rejection of `\(H_0\)` in these tests suggests that volatility is time varying and predictable. This leads to an ARCH model of volatility. --- * **Example** `$$\Delta GOOG_t=\underset{(5\times10^{-4})}{7\times 10^{-4}} + \underset{(0.0316)}{0.0389}\Delta GOOG_{t-1}+\epsilon_t$$` -- Auxiliary regression `$$\hat\epsilon^2_t=\underset{(3\times10^{-05})}{2\times10^{-04}}+\underset{(0.03166)}{0.04372}\hat\epsilon^2_{t-1}+u_t$$` `$$R^2=0.001911\ \ \ \ \ \ \ T=998$$` -- `$$LM=R^2\cdot T{\sim}\chi^2_1\text{ under }H_0$$` `$$LM_{calc}=R^2\cdot T=0.001911\times998=1.91<3.84=LM_{crit}$$` Do not reject `\(H_0\)` --- ##ARCH Models `$$r_t=E[r_t|\mathcal{F}_{t-1}]+\epsilon_t$$` __ARCH models__ specify `$$\epsilon_t=u_t\sigma_t$$` where `\(u_t\sim i.i.d.(0,1)\)` and `\(\sigma^2_t=V(\epsilon_t|\mathcal{F}_{t-1})=g(\epsilon_{t-1}^2,\epsilon_{t-1}^2,\ldots)\)` Such models are referred to as being __conditionally heteroskedastic__. -- The __news process__ `\(\{\epsilon_t\}\)` is driven by underlying __shocks__ `\(u_t\)`, and it is common to assume that `\(u_t\sim i.i.d.N(0,1)\)` -- Implies `$$\frac{\epsilon_t}{\sigma_t}\overset{iid}\sim N(0,1)\ \ \ \ \ \ \ \ \ \ \ \epsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma_t^2)$$` `\(\left\{u_t=\frac{\epsilon_t}{\sigma_t}\right\}\)` is often called the __standardised news__, and `\(\frac{\hat\epsilon_t}{\hat\sigma_t}\)` the __standardised residual__ --- Often don't care about `\(E[r_t|\mathcal{F}_{t-1}]\)` Often take `$$\begin{aligned}r_t&=\epsilon_t\\ &=u_t\sigma_t\\ &\equiv u_t\sqrt{g(\epsilon^2_{t-1}, \epsilon^2_{t-1}, \ldots, \epsilon^2_{t-k})} \end{aligned}$$` -- * **Common ARCH Model** An ARCH(q) model assumes that `\(E(\epsilon_t|\mathcal{F}_{t-1})=0\)` `$$\sigma^2_t=V(\epsilon_t|\mathcal{F}_{t-1})=\alpha_0+\alpha_1\epsilon^2_{t-1}+\alpha_2\epsilon^2_{t-1}+\ldots+\alpha_q\epsilon_{t-1}^2$$` Assume `\(\alpha_i>0\)` to ensure `\(\sigma^2_t>0\)` Assume `\(\sum^q_{i=1}\alpha_i<1\)` to ensure the unconditional variance is well defined. Persistence is often measured by `\(\sum^q_{i=1}\alpha_i<1\)`. -- If `\(\alpha_1=\ldots=\alpha_q=0\)` then `\(\sigma^2_t=\alpha_0\)` and the model reverts to a standard constant volatility model. The conditional variance ($\sigma^2_t$) is often called `\(h_t\)`. --- * **Properties of datagenerated by an ARCH(1) model** `\(\epsilon_t=u_t\sigma_t\)` `\(u_t\sim i.i.d.(0,1)\)` `$$\sigma^2_t=\alpha_0 +\alpha_1 \epsilon_{t-1}^2$$` This (non-constant) conditional variance model generates __volatility clustering__. <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-8-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> --- * **ARCH and Stylized Facts** 1. Heavy tails (positive excess kurtosis) 2. Asymmetry (negative skewness) 3. Lack of persistent in levels of returns 4. Volatility clustering -- * **Law of Iterated Expectations (L.I.E.)** The order of taking expectations does not matter `$$\mathbb{E}[X_t]=\mathbb{E}[\mathbb{E}[X_t|\mathcal{F}_{t-1}]]$$` * **Law of Total Variance** `$$Var(X_t)=\mathbb{E}[Var(X_t|\mathcal{F}_{t-1})]+Var(\mathbb{E}[X_t|\mathcal{F}_{t-1}])$$` --- ###Unconditional moments of the ARCH(1) model * **Unconditional mean** `$$E(\epsilon_t)=0$$` -- > __Proof__ > By the L.I.E > `$$E(\epsilon_t)=E[E(\epsilon_t|\mathcal{F}_{t-1})]$$` > and since `\(\epsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)\)` it follows that `\(E(\epsilon_t|\mathcal{F}_{t-1})=0\)` --- * **Unconditional covarance** `$$Cov(\epsilon_t, \epsilon_{t-k})=0\forall k\geq1$$` -- > __Proof__ > By definition of covariance (and stationarity of `\({\epsilon_t}\)`) > `$$\begin{aligned}Cov(\epsilon_t, \epsilon_{t-k} ) &=E[(\epsilon_t-E(\epsilon_t))(\epsilon_{t-k}-E(\epsilon_{t-k}))]\\ &=E[\epsilon_t\epsilon_{t-k}] \end{aligned}$$` > We know `\(E(\epsilon_t)=E(\epsilon_{t-k})=0\)`. > By L.I.E. > `$$E[\epsilon_t\epsilon_{t-k}]=E[E[\epsilon_t\epsilon_{t-k}|\mathcal{F}_{t-1}]]=E[\epsilon_{t-k}E[\epsilon_t|\mathcal{F}_{t-1}]]$$` > And `\(E(\epsilon_t|\mathcal{F}_{t-1})=0\)` --- * **Unconditional variance** `$$V(\epsilon_t)=\frac{\alpha_0}{(1-\alpha_1)}$$` -- > __Proof__ > `\(E(\epsilon_t)=0\)` then `\(V(\epsilon_t)=E(\epsilon^2_t)\)` > By L.I.E. > `$$\mathbb{E}[\epsilon_t^2]=\mathbb{E}[\mathbb{E}[\epsilon_t^2|\mathcal{F}_{t-1}]]$$` > `\(\epsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)\)` then `\(\sigma_t^2=V(\epsilon_t|\mathcal{F}_{t-1})=E(\epsilon^2_t|\mathcal{F}_{t-1})\)` > `$$\begin{aligned}V(\epsilon_t)&=E(\epsilon^2_t)=\mathbb{E}[\mathbb{E}[\epsilon_t^2|\mathcal{F}_{t-1}]]\\ &=\sigma_t^2= \alpha_0 +\alpha_1 E(\epsilon_{t-1}^2)\\ &=\alpha_0 +\alpha_1 V(\epsilon_{t-1})\\ &=\alpha_0 +\alpha_1 V(\epsilon_{t})\ \ (stationary)\\ \Rightarrow V(\epsilon_t)&=\frac{\alpha_0}{(1-\alpha_1)}\end{aligned}$$` --- * **Unconditional third moment** If we assume `\(u_i\overset{iid}{\sim}N(0,1)\)` then `$$E(\epsilon^3_t)=0$$` -- > __Proof__ > By L.I.E. > `$$\mathbb{E}[\epsilon_t^3]=\mathbb{E}[\mathbb{E}[\epsilon_t^3|\mathcal{F}_{t-1}]]$$` > `$$\begin{aligned}E(\epsilon^3_t)&={E}[{E}[\sigma_t^3u_t^3|\mathcal{F}_{t-1}]]\\ &={E}\sigma_t^3[{E}[u_t^3|\mathcal{F}_{t-1}]]\end{aligned}$$` > `\(u_t\sim N(0,1)\)` --- * **Unconditional fourth moment** `$$E(\epsilon^4_t)=\frac{3\alpha_0^2(1+\alpha_1)}{(1-3\alpha^2_1)(1-\alpha_1)}$$` and so __kurtosis__ for `\(\epsilon_t\)` is `$$\kappa=\frac{3(1-\alpha_1^2)}{1-3\alpha^2_1}>3$$` __Lemma__ If `\(X\sim N (0,\sigma^2)\)`, then `$$\mu_{2s}=E\left[(X-\mu)^{2s} \right]=\frac{\sigma^{2s}(2s)!}{2^ss!}$$` so `\(\mu_4=\frac{\sigma^4(4!)}{4\cdot2!}=\frac{\sigma^4(1\times2\times3\times4)}{4(1\times2)}=3\sigma^4\)` -- __Proof__ By L.I.E. `$$\begin{aligned}E(\epsilon^4_t)&=E[E[\epsilon^4_t|\mathcal{F}_{t-1}]]\\ &=E[\sigma^4_tE[u^4_t|\mathcal{F}_{t-1}]] \end{aligned}$$` --- Then, since `\(u_t|\mathcal{F}_{t-1}\sim N(0,1)\)`, from the lemma we have `$$E[u_t^4|\mathcal{F}_{t-1}]=3$$` and so `$$E(\epsilon^4_t)=E[\sigma^4_t\times3]=3E[(\sigma_t^2)^2]$$` -- Since `\(\sigma^2_t=\alpha_0+\alpha_1\epsilon^2_{t-1}\)` `$$\begin{aligned}E(\epsilon_t^4)&=3E[(\alpha_0+\alpha_1\epsilon_{t-1}^2)(\alpha_0+\alpha_1\epsilon^2_{t-1})]\\ &=3E[\alpha_0^2+2\alpha_0\alpha_1\epsilon^2_{t-1}+\alpha_1^2\epsilon^4_{t-1}]\\ &=3\alpha_0^2+6\alpha_0\alpha_1E[\epsilon^2_{t-1}] +3\alpha_1^2E[\epsilon^4_{t-1}] \end{aligned}$$` Since stationary `\(E(\epsilon^4_{t-1})=E(\epsilon^4_{t})\)` and `\(E(\epsilon^2_{t-1})=E(\epsilon^2_{t})=V(\epsilon_t)\)` Substituting in `\(V(\epsilon_t)=\frac{\alpha_0}{1-\alpha_1}\)` `$$\begin{aligned} E(\epsilon^4_t)&=3\alpha_0^2+\frac{6\alpha_0^2\alpha_1}{(1-\alpha_1)}+3\alpha_1^2E(\epsilon^4_t)\\ &=\frac{3\alpha_0^2(1+\alpha_1)}{(1-3\alpha^2_1)(1-\alpha_1)} \end{aligned}$$` --- * **Kurtosis** `$$\begin{aligned}\kappa&=\frac{E(\epsilon^4_t)}{[E(\epsilon^2_t)]^2}=\frac{[3\alpha_0^2(1+\alpha_1)]}{(1-3\alpha_1^2)(1-\alpha_1)}\times\frac{(1-\alpha_1)^2}{\alpha_0^2}\\ &=\frac{[3(1+\alpha_1)]}{(1-3\alpha_1^2)}\times\frac{(1-\alpha_1)}{1}\\ &=3\times\frac{(1-\alpha^2_1)}{(1-3\alpha^2_1)}\end{aligned}$$` Must be that `\(\kappa>0\)`. Requires `\(3\alpha^2_1<1\)` for fourth moment to exist. For$0<\alpha_1<\sqrt{1/3}$ `$$(1-\alpha_1^2)>(1-3\alpha^2_1)$$` `$$\frac{(1-\alpha^2_1)}{(1-3\alpha^2_1)}>1$$` `$$\kappa=3\times\frac{(1-\alpha^2_1)}{(1-3\alpha^2_1)}>3$$` Not Normal --- The unconditional distribution of the `\({\epsilon_t}\)` from an ARCH model __cannot be a normal distribution__ because there is __too much kurtosis__. - Returns are not normal - Tails are too thick - Can use a conditional normal random variable to model returns -- ARCH(1) Model allows us to capture two important stylized facts of returns data: (1) Returns have thick tails, i.e. not normal (2) Risk (or volatility) is time-varying and autoregressive, i.e. returns display ARCH-like behavior. --- ###News Impact Curve The __news impact curve (NIC)__ is a plot of `\(\sigma_t^2\)` (vertical) against `\(\epsilon_{t-1}\)` (horizontal), holding all else (in the past) constant. NIC plot summarises how the current volatility is influenced by the __last period's news__, according to the model. <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-9-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> --- `\(\sigma^2_t\)` is a __nondecreasing__ function of the __magnitude__ of past news (i.e. `\(|\epsilon_{t-1}|\)`) The NIC for the ARCH(1) is __symmetric__ about `\(\epsilon_{t-1}=0\)`, indicating that the sign of the news does not matter. (not the resonable) -- `\(\alpha_1\)` determines the extent to which past news is reflected in current volatility. `\(\alpha_0\)` determines the __vertical position__ of the NIC --- ###Estimation * **Simple Estimation of ARCH - OLS** `$$E[r_t^2|\mathcal{F}_{t-1}]=\alpha_0 +\alpha_1r^2_{t-1}$$` Define `\(y_t=r_t^2\)` `$$y_t=\alpha_0+\alpha_1y_{t-1} + v_t$$` `$$\underset{\alpha_0, \alpha_1}{\min}\sum^T_{t=2}(y_t-\alpha_0-\alpha_1y_{t-1})^2$$` --- * **Estimation of an ARCH model - GLS** OLS does not provide efficient estimators. Can correct using GLS. 1. Obtain OLS estimates `\(\hat\alpha_0,\hat\alpha_1\)` 2. Compute `\(f_t=\hat a_0+\hat a_1y_{t-1}\)` -- 3. Regress `\([(y_t/f_t)-1]\)` on `\(1/f_t\)` and `\((y_{t-1}/f_t)\)` (to obtain `\(\bar a_0, \bar a_1\)`) 4. The GLS estimator is given by `$$\left(\begin{array}{c}\hat{\hat a}_0\\ \hat{\hat a}_1\end{array}\right)=\left(\begin{array}{c}\hat a_0+\bar a_0\\ \hat a_1+\bar a_1\end{array}\right)$$` --- * **Estimation of an ARCH model - MLE** `$$LL=-\frac T2\ln(2\pi)-\frac12\sum^T_{t=1}\ln\sigma^2_t-\frac12\sum^T_{t=1}\frac{(r_t-c-\phi_1r_{t-1})^2}{\sigma^2_t}$$` with `$$\sigma^2_t=\alpha_0+\alpha_1(r_{t-1}-c-\phi_1r_{t-2})^2$$` The form of this LL comes from the AR(1)-ARCH(1) specification and the assumption that `\(u_t\overset{iid}\sim N(0,1)\)` -- We need to choose `\(c\)`, `\(\phi_1\)`, `\(\alpha_0\)` and `\(\alpha_1\)` to maximise the LL. Our GLS estimator is asymptotically equivalent to MLE based on Gaussuan errors. --- ###Forecasting: an AR(1)-ARCH(1) model Fitted model `$$r_t=\underset{(0.0003919)}{0.001}+\underset{(0.03712)}{0.01402}r_{t-1}+\epsilon_t$$` with `$$\epsilon_t=\sigma_tu_t$$` `$$\sigma^2_t=\underset{(0.000009047)}{0.000137}+\underset{(0.07686)}{0.4478}\epsilon^2_{t-1}$$` `\(N=999\)` daily frequency -- Forecast next day `\(r_{1000}\)` We have `\(r_{999}=0.005064\)`, `\(P_{999}=813.67\)`, `\(\hat\epsilon_{999}=0.0040430193\)` --- `\(r_{999}=0.005064\)`, `\(P_{999}=813.67\)`, `\(\hat\epsilon_{999}=0.004043\)` -- Conditional mean: `$$\begin{aligned}\hat E[r_{1000}|\mathcal{f}_{999}]&=0.001+0.01402\times0.005064\\ &=0.001070997 \end{aligned}$$` -- Conditional variance: `$$\begin{aligned}\hat V[r_{1000}|\mathcal{F}_{999}]&=\hat V[\epsilon_{1000}|\mathcal{F}_{999}]\\ &=0.000137+0.4478\times0.004043\\ &=0.001947455 \end{aligned}$$` -- Volatility: `$$\hat\sigma_{1000|\mathcal{F}_{999}}=\sqrt{0.001947455}=0.04412998$$` --- `\(u_t\sim N(0,1)\)`, `\(\epsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)\)` `\(\Rightarrow r_t\sim N\)` -- Prediction interval: `$$0.001070997\pm1.96\times0.04412998$$` `$$[-0.08542376,0.08756576]$$` Note that our prediction interval would have been wider if we had not used our ARCH model for predicting variance. -- Prediction interval for `\(P_1000\)`: `$$[813.67e^{-0.08542376},\ 813.67e^{0.08756576}]$$` --- **Diagnostics for an ARCH model** Based on standardized residuals `\(\hat u_t=\frac{\hat \epsilon_t}{\hat\sigma_t}\)` - ACF plot - PACF plot - Simple test - BG test -- - ARCH LM test to determine if serial correlation in `\(\hat u_t^2\)` remains If it does, include more ARCH terms. --- * **ARCH(q) Model** `\(\epsilon_t=u_t\sigma_t\)`, `\(u_t\sim i.i.d.N(0,1)\)` `$$\sigma^2_t=\alpha_0+\alpha_1\epsilon^2_{t-1}\ldots+\alpha_q\epsilon^2_{t-q}$$` This model generates volatility clustering patterns and news impact curves that resemble those for ARCH(1) models. -- * Problems with ARCH modelling Im many cases, we need `\(q\)` to be very large. - Can be fixed by GARCH The `\(\hat u_t\)` do not appear to have come from a normally distributed `\(u_t\)` - Can be fixed by changing the assumptions about `\(u_t\)` (e.g. t-distribution) and with a different likelihood --- ##GARCH Models ARCH models have difficulty modelling the autocorrelation in the squared residuals. GARCH models generalise ARCH models to include lags of `\(\sigma^2_t\)` in the conditional variance equation as well as lags of `\(\epsilon^2_t\)`, and this often alleviates this difficulty. -- GARCH(p,q) `$$\sigma^2_t=\alpha_0+\alpha_1\epsilon^2_{t-1}+\ldots+\alpha_q\epsilon^2_{t-q}\ \ \ +\beta_1\sigma^2_{t-1}+\ldots+\beta_p\sigma^2_{t-p}$$` Assume all `\(\alpha_i>0\)` and `\(\beta_j>0\)` to ensure that `\(\sigma^2_t>0\)` Assume `\(\sum^{i=q}_{i=1}\alpha_i+\sum^{j=p}_{j=1}\beta_j<1\)` to ensure that `\(V(\epsilon_t)>0\)` -- The `\(\alpha_i\)` and `\(\beta_j\)` determine how past news affects current volatility and this persistence is often measured by `\(\gamma=\sum^{i=q}_{i=1}\alpha_i+\sum^{j=p}_{j=1}\beta_j\)` If `\(\alpha_0\)` is the only non-zero parameter then we have constant volatility. --- * **Conditional moments of GARCH(p,q) errors** `\(\epsilon_t=u_t\sigma_t\)`, `\(u_t\sim N(0,1)\)`, `\(\frac{\epsilon_t}{\sigma_t}\sim N(0,1)\)`, `\(\epsilon_t|\mathcal{F}_{t-1}\sim N（0,\sigma^2_t)\)` - `$$E(\epsilon_t|\mathcal{F}_{t-1})=0$$` - `$$V(\epsilon_t|\mathcal{F}_{t-1})=\sigma_t^2=\alpha_0+\sum^q_{i=1}\beta_j\sigma^2_{t-j}$$` - `$$Cov(\epsilon_t,\epsilon_{t-k}|\mathcal{F}_{t-1})=0\forall k >1$$` - The skewness of `\(\epsilon_t|\mathcal{F}_{t-1}\)` is zero - The kurtosis of `\(\epsilon_t|\mathcal{F}_{t-1}\)` is 3 -- `$$Cov(\epsilon_t,\epsilon_{t-k}|\mathcal{F}_{t-1})=E(\epsilon_t,\epsilon_{t-k}|\mathcal{F}_{t-1})=\epsilon_{t-k}E(\epsilon_t|\mathcal{F}_{t-1})=0$$` --- ###Unconditional moments of the GARCH(p,q) errors * **Unconditional mean** `$$E(\epsilon_t)=0$$` -- > __Proof__ > By the L.I.E > `$$E(\epsilon_t)=E[E(\epsilon_t|\mathcal{F}_{t-1})]$$` > and since `\(\epsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)\)` it follows that `\(E(\epsilon_t|\mathcal{F}_{t-1})=0\)` --- * **Unconditional covarance** `$$Cov(\epsilon_t, \epsilon_{t-k})=0\forall k\geq1$$` -- > __Proof__ > By definition of covariance (and stationarity of `\({\epsilon_t}\)`) > `$$\begin{aligned}Cov(\epsilon_t, \epsilon_{t-k} ) &=E[(\epsilon_t-E(\epsilon_t))(\epsilon_{t-k}-E(\epsilon_{t-k}))]\\ &=E[\epsilon_t\epsilon_{t-k}] \end{aligned}$$` > We know `\(E(\epsilon_t)=E(\epsilon_{t-k})=0\)`. > By L.I.E. > `$$E[\epsilon_t\epsilon_{t-k}]=E[E[\epsilon_t\epsilon_{t-k}|\mathcal{F}_{t-1}]]=E[\epsilon_{t-k}E[\epsilon_t|\mathcal{F}_{t-1}]]$$` > And `\(E(\epsilon_t|\mathcal{F}_{t-1})=0\)` --- * **Unconditional variance** `$$V(\epsilon_t)=\frac{\alpha_0}{(1-\sum^{i=q}_{i=1}\alpha_i-\sum^{j=p}_{j=1}\beta_j)}$$` -- > __Proof__ > `\(E(\epsilon_t)=0\)` then `\(V(\epsilon_t)=E(\epsilon^2_t)\)` > By L.I.E. > `$$\mathbb{E}[\epsilon_t^2]=\mathbb{E}[\mathbb{E}[\epsilon_t^2|\mathcal{F}_{t-1}]]$$` > `\(\epsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)\)` then `\(\sigma_t^2=V(\epsilon_t|\mathcal{F}_{t-1})=E(\epsilon^2_t|\mathcal{F}_{t-1})\)` > `$$\begin{aligned}V(\epsilon_t)&=E(\epsilon^2_t)=\mathbb{E}[\mathbb{E}[\epsilon_t^2|\mathcal{F}_{t-1}]]\\ &=\sigma_t^2= {\alpha_0}+{\sum^{i=q}_{i=1}\alpha_iE(\epsilon^2_{t-i})+\sum^{j=p}_{j=1}\beta_jE(\sigma^2_{t-j})}\\ &=\alpha_0 + V(\epsilon_{t})\left[{\sum^{i=q}_{i=1}\alpha_i+\sum^{j=p}_{j=1}\beta_j}\right]\ \ (stationary)\\ \Rightarrow V(\epsilon_t)&=\frac{\alpha_0}{(1-\sum^{i=q}_{i=1}\alpha_i-\sum^{j=p}_{j=1}\beta_j)}\end{aligned}$$` --- * **Unconditional third moment** If we assume `\(u_i\overset{iid}{\sim}N(0,1)\)` then `$$E(\epsilon^3_t)=0$$` -- > __Proof__ > By L.I.E. > `$$\mathbb{E}[\epsilon_t^3]=\mathbb{E}[\mathbb{E}[\epsilon_t^3|\mathcal{F}_{t-1}]]$$` > `$$\begin{aligned}E(\epsilon^3_t)&={E}[{E}[\sigma_t^3u_t^3|\mathcal{F}_{t-1}]]\\ &={E}\sigma_t^3[{E}[u_t^3|\mathcal{F}_{t-1}]]\end{aligned}$$` > `\(u_t\sim N(0,1)\)` --- * **Kurtosis** It can be shown that kurtosis `\(\kappa>3\)` (proof not supplied) Should be true since when `\(\beta=0\)` we get ARCH(1) which has `\(\kappa>3\)` -- Conditions needed to ensure `\(\sigma^2_t>0\)` - `\(\alpha_0>0\)` - `\(\alpha_i\ge0\)` - `\(\beta_j\ge0\)` -- For GARCH(1,1) `$$\epsilon_t=\sigma_tu_t,\ \ \ \sigma^2_t=\alpha_0+\alpha_1\epsilon^2_{t-1}+\beta_1\sigma_{t-1}^2$$` Conditions needed to ensure `\(\sigma^2_t>0\)` - `\(\alpha_0>0\)` - `\(\alpha_1\ge0\)` - `\(\beta_1\ge0\)` --- * **Connection to AR and ARMA** ARCH(p) was an AR(p) in squares Can view GARCH(p,q) as ARMA(p,q) in squares -- For GARCH(1,1) model: Assuming `\(E[\epsilon_t^4]<\infty\)` - Define `\(\eta_t=\sigma^2_t(u^2_t-1)\)` - Define `\(y_t=\epsilon^2_t\)` - Then, `\(y_t=\alpha_0+(\alpha_1+\beta_1)y_{t-1}-\beta_1\eta_{t-1}+\eta_t\)` -- > __Proof__ > `\(y_t-\eta_t=\sigma^2_t\)` > `$$\begin{aligned}y_t-\eta_t&=\alpha_0+\alpha_1y_{t-1}+\beta_1\sigma^2_{t-1}\\ &=\alpha_0 +\alpha_1y_{t-1}+\beta_1(y_{t-1}-\eta_{t-1})\\ & =\alpha_0+(\alpha_1+\beta_1)y_{t-1}-\beta_1\eta_{t-1} \end{aligned}$$` --- ###News Impact curves for GARCH models The __news impact curve (NIC)__ is a plot of `\(\sigma_t^2\)` (vertical) against `\(\epsilon_{t-1}\)` (horizontal), holding all else (in the past) constant. For a GARCH(1,1) model the NIC is given by `$$NIC(\epsilon_{t-1})=\alpha_0+\frac{\beta_1\alpha_0}{(1-\alpha_1-\beta_1)}+\alpha_1\epsilon^2_{t-1}$$` <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-12-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> `\(\hat\alpha_1\ll\hat\beta_1\)` is typical of GARCH(1,1) models --- * **Multi-step forecasting of GARCH(1,1) volatility** `$$\begin{aligned}E(\sigma^2_{t+1})&=\alpha_0+\alpha_1(\hat\epsilon_1)^2+\beta_1\sigma^2_t\\ E(\sigma^2_{t+2})&=\alpha_0+\alpha_1E(\epsilon_{t+1}^2)+\beta_1E(\sigma^2_{t+1})\\ &= \alpha_0+\alpha_1E(u_{t+1}^2)E(\sigma_{t+1}^2)+\beta_1E(\sigma^2_{t+1})\\ &=\alpha_0+(\alpha_1+\beta_1)E(\sigma^2_{t+1})\\ E(\sigma^2_{t+3})&= \alpha_0+(\alpha_1+\beta_1)E(\sigma^2_{t+2})\\ &= \alpha_0+(\alpha_1+\beta_1)[\alpha_0+(\alpha_1+\beta_1)E(\sigma^2_{t+1})]\\ &= \alpha_0[1+(\alpha_1+\beta_1)](\alpha_1+\beta_1)^2E(\sigma^2_{t+1})\\ &\vdots\\ E(\hat\sigma^2_{t+k})&=\frac{\alpha_0\left[1-(\alpha_1+\beta_1)^{k-1}\right]}{1-(\alpha_1+\beta_1)}+(\alpha_1+\beta_1)^{k-1}E(\sigma^2_{t+1})\\ &\rightarrow \frac{\alpha_0}{1-(\alpha_1+\beta_1)}\text{ as }k\rightarrow \infty\ ( \text{i.e. unconditional }\sigma^2_\epsilon) \end{aligned}$$` --- ###Comparing ARCH/GARCH Models * **Likelihood Ratio test** ARCH(p) models are nested in GARCH(p,q) models - `\(\beta_1=\cdots=\beta_q=0\Longrightarrow\)` ARCH(p) - Can jointly test for significance of `\(\beta_j\)`s -- If the ARCH specification in question is nested in the GARCH model, can use `$$\text{LR}=2\times\left(\text{LL}(\hat\theta_G)-\text{LL}(\hat\theta_A)\right)\overset{asy}{\sim}\chi^2_q$$` LL$(\hat\theta_G)$: likelihood from GARCH model LL$(\hat\theta_A)$: likelihood from ARCH model --- ##Asymmetric GARCH models Standard ARCH/GARCH models hace symmetric impacts of good and bad news -- * **Leverage effect** Leverage is the ratio of a firm's debt to its equity `\(L=D/E\)` A good shock `\(\epsilon_t\Rightarrow P\uparrow\Rightarrow E\uparrow\Rightarrow L\downarrow\Rightarrow\)` lower risk A bad shock `\(\epsilon_t\Rightarrow P\downarrow\Rightarrow E\downarrow\Rightarrow L\uparrow\Rightarrow\)` higher risk -- A bad shock of size `\(s\)` has a bigger % effect than a good shock of size `\(s\)` because `\(s\)` contrubutes to the denominator of `\(L\)` -- - Exchange: Uncertainties that arise from appreciation are less than those arising from depreciation - Economy might react more to + inflation shocks (than - shocks) - A rise in interest rats might have greater effect than a fall --- * **Test for Volatility Asymmetry** Can be tested using LM tests based on the squard standardised residuals `\(\hat u_t^2\)` `$$H_0:\text{Volatility is symmetric with respect to positive or negative shocks}$$` Define dummies `\(S^+_t\)` and `\(S^-_t\)` by `\(S^+_t=1\)` if `\(\epsilon_t\ge1\)` and `\(S^-_t=1\)` if `\(\epsilon_t\le1\)` -- Auxilliary regression `$$\hat u^2_t=\phi_0+\phi_1S^-_{t-1}+\xi_t$$` `$$T\cdot R^2\sim\chi^2_1\text{ under } H_0$$` Prejection of `\(H_0\)` would suggest that we need a new model that gives rise to an asymmetric news impact curve. --- Other LM type tests include - `\(\hat u_t^2=\phi_0+\phi_1S^-_{t-1}\epsilon_{t-1}+\xi_t\)` (negative size test) `\(\phi_1<0\Rightarrow\)` neg shocks raise volatility by more and how much more depends on the size of the shocks (t-test on `\(\hat\phi_1\)`) -- - `\(\hat u_t^2=\phi_0+\phi_1S^+_{t-1}\epsilon_{t-1}+\xi_t\)` (positive size test) `\(\phi_1>0\Rightarrow\)` pos shocks raise volatility by more and how much more depends on the size of the shocks (t-test on `\(\hat\phi_1\)`) --- Both `$$T\cdot R^2\sim\chi^2_1$$` --- Joint null hypothesis `$$\hat u_t^2=\phi_0+\phi_1S^-_{t-1}+\phi_2S^-_{t-1}\epsilon_{t-1}+\phi_3S^+_{t-1}\epsilon_{t-1}+\xi_t$$` `$$H_0:\text{ no volatility asymmetry }$$` `$$T\cdot R^2\sim\chi^2_3$$` -- There is strong evidence of volatility asymmetry in stock markets, but less evidence of this in other asset markets. --- * **Exponential GARCH or EGARCH model** `$$\ln\sigma^2_t=\alpha_0+\alpha_1(|u_{t-1}|-E[|u_{t-1}|])+\gamma u_{t-1}+\beta_1\ln\sigma^2_{t-1}$$` -- If `\(\gamma\)` is negative, bad news has a larger impact than good news, and negative `\(u_{t-1}\)` will increase volatility. We use logs, so that we avoid the restriction that `\(\alpha_i,\beta_i\ge0\)` --- ###News impact curves for various asymmetry model * **GJR-GARCH model** `$$\sigma^2_t=\alpha_0+\alpha_1\epsilon^2_{t-1}+\alpha^-_1(S^-_1\epsilon^2_{t-1})+\beta\sigma^2_{t-1}$$` NIC `$$\sigma^2_t=\left\{\begin{array}{ll}A+\alpha_1\epsilon^2_{t-1}& \text{for } \epsilon_{t-1}>0\\ A+(\alpha_1+\alpha_1^-)\epsilon^2_{t-1}& \text{else}\end{array} \right.$$` `$$A=\alpha_0+\beta\sigma^2$$` `$$\sigma^2=\alpha_0/[1-(\alpha_1+\alpha_1^-/2)-\beta_1]$$` In this case of GJR-GARCH model the curve has its minimum at `\(\epsilon_{t-1}\)` --- <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-13-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> --- * **EGARCH model** `$$\ln\sigma^2_t=\alpha_0+\alpha_1(|u_{t-1}|-E[|u_{t-1}|])+\gamma u_{t-1}+\beta_1\ln\sigma^2_{t-1}$$` `\(u_{t-1}=\epsilon_t/\sigma_t\)` NIC `$$\sigma^2_t=\left\{\begin{array}{ll}A\exp\left[\frac{\gamma+\alpha_1}{\sigma}\epsilon^2_{t-1}\right]& \text{for } \epsilon_{t-1}>0\\ A\exp\left[\frac{\gamma-\alpha_1}{\sigma}\epsilon^2_{t-1}\right]& \text{else}\end{array} \right.$$` `$$A(=\sigma^2)^{\beta_1}\exp[\alpha_0]$$` `$$\sigma^2=\exp\left([\alpha_0-\alpha_1\sqrt{2/\pi}]/(1-\beta_1)\right)\text{ if }u_t\sim N(0,1)$$` --- <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-14-1.png" width="45%" height="45%" style="display: block; margin: auto;" /> -- <img src="ETC3460_slides_S1_2018_files/figure-html/unnamed-chunk-15-1.png" width="45%" height="45%" style="display: block; margin: auto;" />