Seasonality

You are interested in predicting occupancy levels for a resort hotel chain and have obtained the chain’s quarterly occupancy levels for the most recent 40 quarters (10 years). You decide to model the quarterly occupancy time series using the AR(1) model:

Determine whether seasonality exists using the results presented in the following example.

Autoregression Output for Log Quarterly Hotel Occupancy

Solution

The bottom part of the table contains the residual autocorrelations for the first four lags of the time series. What stands out is the relatively large autocorrelation and t statistic for the fourth lag. With 39 observations and two parameters, ($b_0$ and $b_1$), there are 37 degrees of freedom.

At a significance level of 5%, the critical t value is 2.026.

The t statistics indicate that none of the first three lagged autocorrelations is significantly different from zero. However, the t statistic at Lag 4 is 5.4460, which means that we must reject the null hypothesis that the Lag 4 autocorrelation is zero and conclude that seasonality is present in the time series. Thus, we conclude that this model is misspecified and will be unreliable for forecasting purposes.

We need to include a seasonality term to make the model more correctly specified.

Correcting for seasonality. The interpretation of seasonality in the previous example is that occupancy in any quarter is related to occupancy in the previous quarter and the same quarter in the previous year. For example, fourth quarter 2022 occupancy is related to third quarter 2022 occupancy as well as fourth quarter 2021 occupancy.

To adjust for seasonality in an AR model, an additional lag of the dependent variable (corresponding to the same period in the previous year) is added to the original model as another independent variable. For example, if quarterly data are used, the seasonal lag is 4; if monthly data are used the seasonal lag is 12; and so on.

We continue with our resort occupancy level example, where the significant residual correlation at Lag 4 indicates seasonality in the quarterly time series. By testing the correlations of the error terms, it appears that occupancy levels in each quarter are related not only to the previous quarter, but also to the corresponding quarter in the previous year. To adjust for this problem, we add a lagged value of the dependent variable to the original model that corresponds to the seasonal pattern.

To model the autocorrelation of the same quarters from year to year, we use an AR(1) model with a seasonal lag: ln $x_t$ = $b_0$ + $b_1$(ln $x_{t-1}$) + $b_2$(ln $x_{t-4}$) + ${\epsilon }_t$. Note that this specification, the inclusion of a seasonal lag, does not result in an AR(2) model. It results in an AR(1) model incorporating a seasonal lag term.

The results obtained when this model is fit to the natural logarithm of the time series are presented in the following. Determine whether the model is specified correctly.

Solution

Notice in the bottom of the table that the fourth lag residual autocorrelation has dropped substantially and is, in fact, no longer statistically significant. Also notable in these results is the improvement in the R square for the adjusted model (94.9%) compared to the R square from the original model (79.3%). The results shown in the figure indicate that, by incorporating a seasonal lag term, the model is now specified correctly.

Based on the regression results from the previous example and the occupancy levels over the past year (presented next), forecast the level of hotel occupancy for the first quarter of 2023.

Quarterly Hotel Occupancy Levels

Solution

We express the seasonally adjusted forecasting equation as:

To forecast the occupancy level for the hotel chain for the first quarter of 2023 (i.e., 2023.1), the following computation is made:

The forecasted level of hotel occupancy for the first quarter of 2023 is 603,379, a significant increase over the same quarter the previous year.