92 Homework - MLE and Bayesian inference in the AR(1)

Caution

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--- date: 2024-10-23 title: "Homework - MLE and Bayesian inference in the AR(1)" subtitle: Time Series Analysis description: "This lesson we will define the AR(1) process, Stationarity, ACF, PACF, differencing, smoothing" categories: - Bayesian Statistics keywords: - time series - stationarity - strong stationarity - weak stationarity - lag - autocorrelation function (ACF) - partial autocorrelation function (PACF) - smoothing - trend - seasonality - differencing operator - back shift operator - moving average --- ::::: {.content-visible unless-profile="HC"} ::: {.callout-caution} Section omitted to comply with the Honor Code ::: ::::: ::::: {.content-hidden unless-profile="HC"} 1. Consider an autoregressive model given by $y_t = \phi y_{t-1} + \epsilon_t, \quad \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)$, The maximum likelihood estimator of $\phi$ based on $T$ observations and using the full likelihood can be found by maximizing the expression: - [ ] $-\sum_{t=2}^{T} (y_t - \phi y_{t-1})^2$ - [ ] $\sum_{t=2}^{T} (y_t - \phi y_{t-1})^2$ - [x] $\log(1-\phi^2) - y_1^2(1-\phi^2) - \sum_{t=2}^T(y_t - \phi y_{t-1})^2$ - [ ] $- y_1^2(1-\phi^2) - \sum_{t=2}^T(y_t - \phi y_{t-1})^2$ 2. Consider an autoregressive process of the form - [ ] $\hat{v}_{MLE} = \frac{\sum_{t=2}^T (y_t - \hat{\phi}_{MLE} y_{t-1})^2}{(T-2)}$ - [x] $\hat{v}_{MLE} = \frac{\sum_{t=2}^T (y_t - \hat{\phi}_{MLE} y_{t-1})^2}{T}$ - [ ] None of the above - [ ] $\hat{v}_{MLE} = \frac{\sum_{t=2}^T (y_t - \hat{\phi}_{MLE} y_{t-1})^2}{(T-1)}$ 3. Mark all the true statements below: - [ ] Consider and AR(1) with parameters $\phi$ and $\nu$. The maximum likelihood estimator for $\nu$ obtained using the full likelihood and $T$ observations is an unbiased estimator. - Consider and AR(1) with parameters $\phi$ and $\nu$. The posterior mode for $\phi$ obtained assuming a reference prior, the full likelihood, and based on $T$ observations, is equal to the maximum likelihood estimator for $\phi$ under the full likelihood. - [ ] Consider and AR(1) with parameters $\phi$ and $\nu$. The maximum likelihood estimator for $\nu$ obtained using the conditional likelihood and $T$ observations is an unbiased estimator. - [x] Consider and AR(1) with parameters $\phi$ and $\nu$. The posterior mode for $\phi$ obtained assuming a reference prior, the conditional likelihood, and based on $T$ observations, is equal to the maximum likelihood estimator for $\phi$ under the conditional likelihood. 4. Mark all the true statements below: - [ ] Consider and AR(1) with parameters $\phi$ and $\nu$. Maximum likelihood estimation for $\phi$ under the conditional likelihood is available in closed form, i.e., we can write down the MLE explicitly as a function of $y_{1:T}$ without requiring a numerical optimization method to obtain this MLE. - [x] Consider and AR(1) with parameters $\phi$ and $\nu$. Maximum likelihood estimation for $\phi$ under the conditional likelihood is not available in closed form, i.e., we cannot write down the MLE explicitly as a function of $y_{1:T}$ without requiring a numerical optimization method to obtain this MLE. - [x] Consider and AR(1) with parameters $\phi$ and $\nu$. Posterior inference under a reference prior and the conditional likelihood can be done using direct sampling by first sampling $\nu$ from an inverse-gamma distribution and then sampling $\phi$ conditional on $\nu$ from a normal distribution. :::::