Review

by Professor Throckmorton
for Time Series Econometrics
W&M ECON 408
Slides

Stationarity¶

Conditions¶

Define covariance stationarity, i.e., what are the conditions for a time series to be weakly stationary?

Random Walk¶

Write down a random walk and solve for its variance, i.e., $Var(y_t)$. Given your answer, is a random walk stationary? Why or why not?

MA Model¶

Invertibility¶

Consider the MA(1) process: $y_t = \varepsilon_t + 0.6\varepsilon_{t-1}$. Is this process invertible? Justify your answer.

Autocovariance¶

Write down an MA(2) model. What is its first autocovariance, $\gamma(1)$?

AR Model¶

Causality¶

Show that $y_t = 0.7 y_{t-1} + \varepsilon_t$ is causal.

ARMA Model¶

ARMA($1,1$) $\rightarrow$ $AR(\infty)$¶

Show that an ARMA($1,1$) process can be rewritten as an AR($\infty$). Find the first three AR coefficients.

Variance¶

Find the variance of an ARMA(1,1) process.

ARIMA Model¶

Differencing¶

Suppose you have the ARIMA($1,1,0$) model $\Delta y_t = 0.5 \Delta y_{t-1} + \varepsilon_t$. Rewrite it in its original (non-differenced) form.

Integration¶

Explain how you would determine the order of integration $d$ for a time series.