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The following guidelines are used to determine the most appropriate model depending on the need:
4. On the other hand, serial correlation in the residuals calls for a more complicated model such as an autoregressive model. If the data has serial correlation, recheck it for covariance stationary before running the AR model. Transform the nonstationary data in the following ways:
5. Model the transformed time series with an autoregressive model, AR (1) model.
6. Test if residuals have autoregressive conditional heteroskedasticity (ARCH).
7. Test the forecasting performance of the out-of-sample forecasting performance and compare this to the in-sample performance of the RMSE.
Question
Consider an AR(1) model used to forecast quarterly retail sales of a certain company based on 200 observations.
$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}+\epsilon_{\text{t}}$$
The residual autocorrelations relating to a certain year are as presented in the following table:
$$\small{\begin{array}{c|c} \textbf{Lag} & \textbf{Autocorrelation} \\ \hline1 & -0.0832 \\ \hline2 & -0.0701 \\ \hline3 & 0.0891 \\ \hline4 & 0.9379\\ \end{array}}$$
The most appropriate statement regarding the AR(1) model at the 5% significance level is that:
- the AR(1) model is sufficient.
- the AR(1) model is misspecified and not appropriate for use.
- the AR (1) model’s error terms are not serially correlated, and thus the model is correctly specified.
Solution
The correct answer is B.
To check whether the model is correctly specified at the 5% level of significance,
$$\text{Standard error}=\frac{1}{\sqrt{\text{T}}}=\frac{1}{\sqrt{200}}=0.0707$$
We can compute the t-statistic for lag one as:
$$\text{t}_{\text{Statistic}}=\frac{\text{Residual Autocorrelation}}{\frac{1}{\sqrt{\text{T}}}}$$
$$\text{t}_{\text{Statistic}}=\frac{-0.0832}{0.0707}=-1.1768$$
Calculating the t-statistic for each lag, we obtain:
$$\small{\begin{array}{c|c|c|c} \textbf{Lag} & \textbf{Autocorrelation} & \textbf{Autocorrelation} & \textbf{T-statistic}\\ \hline1 & -0.0832394 & -0.0832 & -1.1768 \\ \hline2 & -0.0701264 & -0.0701 & -0.9915\\ \hline3 & 0.08908331 & 0.0891 & 1.26025 \\ \hline4 & 0.93786911 & 0.9379 & 13.2659\\ \end{array}}$$
There are 200 observations and two parameters, \(\text{b}_{0}\) and \(\text{b}_{1}\) to be estimated. Thus 198 (200-2) degrees of freedom. The critical t-value at the 5% significance level with 198 degrees of freedom is 1.97.
The t-statistics of the first three lagged autocorrelations are less than the critical value at the 5% significance level. We can therefore conclude that none of the first three lagged autocorrelations is significantly different from zero.
However, notice that the t-statistic for the 4^{th} lag autocorrelation is greater than the critical value. We thus reject the null hypothesis that the 4^{th} lag autocorrelation is zero. The conclusion is that there is seasonality in the time series. This implies that the AR(1) model’s error terms are serially correlated, and thus the model is misspecified and not appropriate for use.
Reading 5: Time Series Analysis
LOS 5 (o) Determine an appropriate time-series model to analyze a given investment problem and justify that choice.