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?