Econ 6016: Problem Set 1

*Due in class on Tuesday, February 18th.

Problem 1

Consider the AR(2) process

\begin{eqnarray} y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \epsilon_t. \end{eqnarray}

1. Show that a necessary condition for stationarity is that the coefficients lie inside the triangle \[ \phi_1 + \phi_2 < 1, \quad \phi_2 - \phi_1 < 1, \quad \mbox{and} \; \phi_2 > -1. \]

2. Now re-parameterize the AR(2) model in terms of partial autocorrelations \(\psi_1\) and \(\psi_2\): \[ \phi_1 = \psi_1(1-\psi_2), \quad \phi_2 = \psi_2. \] Show that the AR(2) process is stationary if and only if \(\psi_1 \in (-1,1)\) and \(\psi_2 \in (-1,1)\).

Problem 2

Demonstrate that the first \(p+1\) Yule-Walker equations for the AR(p) process \[ y_t = \sum_{i=1}^p \phi_i y_{t-i} + \epsilon_t \] are

\begin{eqnarray*} \sigma^2_\epsilon &=& \gamma_{yy,0} - \sum_{i=1}^p \phi_i \gamma_{yy,i} \\ 0 &=& \phi_i \gamma_{yy,0} - \gamma_{yy,i} + \sum_{j=1, j \not=i}^p \phi_j \gamma_{yy,|i-j|}, \quad i = 1,\ldots,p \end{eqnarray*}

Calculate the first 3 autocovariances for the AR(3) process of Problem~2.

Problem 3

Download some aggregate time series from the Economic Database (FRED II) maintained by the Federal Reserve Bank of St. Louis: GDP (implicit price deflator), GDP Implicit price deflator, Real Personal Consumption Expenditure, Real Private Nonresidential Fixed Investment.

• Take logs of GDP, Consumption, and Investment. For each series, estimate a model of the form \[ y_t = \beta_1 + \beta_2 t + u_t \] using OLS.
• According to your estimates, what is the annualized average growth rate (in percent) of GDP, consumption, and investment?
• Compare the standard error estimates produced by the software that you are using for the OLS detrending to the asymptotic standard errors derived in class.
• Compute sample autocorrelation functions for the deviations of output, consumption, and investment (the \(\hat{u}_t\)’s) from their deterministic trend.
• Approximate growth rates of these 3 series by \(\ln y_t - \ln y_{t-1}\) and compute the sample autocorrelation function for GDP growth. Compare the results to (i).
• Compute inflation rates as differences of the log GDP deflator. Compute the sample autocorrelation function for inflation.
• Repeat the above analysis for the subsamples ``before 1970", ``between 1970 and 1982", ``after 1982". Did the persistence and the volatility of the series change?
• Repeat the above empirical analysis with four time series from some other industrialized country.

Reading McConnell, Margaret and Gabriel Perez-Quiros (2000): ``Output Fluctuations in the United States: What has changed since the early 1980’s?’’ American Economic Review, 90(5), 1464-76.

Problem 4

Consider the simple model:

\begin{eqnarray} y_t = \rho y_{t-1} + \epsilon_t, \quad \epsilon_t \sim N(0, \sigma_t^2), \quad |\rho| < 1. \end{eqnarray}

Assume that \(\sigma_t^2\) follows the process:

\begin{align} \sigma_t^2 = \bigg\{\begin{array}{ccl}\sigma^2 & : & \mbox{$t$ is even} \\ \alpha\sigma^2 &: & \mbox{$t$ is odd} \end{array} \end{align}

Consider the least squares estimator \[ \hat\rho_{LS} = \left(\sum_{t=2}^T y_{t-1}^2\right)^{-1}\sum_{t=2}^T y_t y_{t-1}. \]

• Is \(\hat\rho_{LS}\) a consistent estimator for \(\rho\)?

• Derive the asymptotic distribution of \(\sqrt{T}(\hat\rho_{LS} - \rho)\) at \(T\rightarrow\infty\). In particular what is the variance, \(\mathbb V_{\hat\rho}\) of this distribution?

• Consider an estimator of the variance which assumes homoskedasticity:

  \begin{eqnarray*} \frac{\frac{1}{T}\sum_{T=2}^T \epsilon_t^2}{\frac{1}{T}\sum_{T=2}^T y_{t-1}^2} \end{eqnarray*}

  What does this quantity converge in probability to? Call is \(\mathbb V^*_{\hat\rho}\).

• What is the relationship between \(\mathbb V_{\hat\rho}\) and \(\mathbb V^*_{\hat\rho}\)?

Problem 5

The Hodrick-Prescott filter (see lecture notes) has been criticized for amplifying the spectrum at certain business cycle frequencies. Consider a real business cycle model that is driven by a total factor productivity process of the form

\begin{equation} y_t = \phi y_{t-1} + \epsilon_t , \quad \epsilon_t \sim WN(0,\sigma^2) \end{equation}

Let \(y_t^*\) be the HP detrended version of this productivity process. Derive and plot the spectrum of \(y_t\) and \(y^*_t\) for \(\phi=0.95\). Can you find any ``spurious’’ cycles in the detrended data. What happens if \(\phi\) decreases to 0.7?