Review 1A

Review 1A#

by Professor Throckmorton
for Time Series Econometrics
W&M ECON 408/PUBP 616
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 staionary? 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#

Stationarity#

Consider the model \(y_t = 0.5 y_{t-1} - 0.3 y_{t-2} + \varepsilon_t\). Is it stationary? Why or why not?

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.

Unit Root Test#

Write down the null and alternative hypotheses of the augmented Dickey-Fuller test (ADF test). You run an ADF test on a time series and obtain a test statistic of -1.8. If the critical value at the 5% level is -2.86, what is your conclusion?

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