Reading 13: Time-Series Analysis

LOS a: Calculate and evaluate the predicted trend value for a time series, modeled as either a linear trend or a log-linear trend, given the estimated trend coefficients.

1. Yolanda Seerveld is an analyst studying the growth of sales of a new restaurant chain called Very Vegan. The increase in the public’s awareness of healthful eating habits has had a very positive effect on Very Vegan’s business. Seerveld has gathered quarterly data for the restaurant’s sales for the past three years. Over the twelve periods, sales grew from $17.2 million in the first quarter to $106.3 million in the last quarter. Because Very Vegan has experienced growth of more than 500% over the three years, the Seerveld suspects an exponential growth model may be more appropriate than a simple linear trend model. However, she begins by estimating the simple linear trend model:

(sales)_t = α + β × (Trend)_t + ε_t

Where the Trend is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

The analyst then estimates the following model:

(natural logarithm of sales)_t = α + β × (Trend)_t + ε_t

Seerveld compares the results based upon the output statistics and conducts two-tailed tests at a 5 percent level of significance. One concern is the possible problem of autocorrelation, and Seerveld makes an assessment based upon the first-order autocorrelation coefficient of the residuals that is listed in each set of output. Another concern is the stationarity of the data. Finally, the analyst composes a forecast based on each equation for the quarter following the end of the sample.

Are either of the slope coefficients statistically significant?

A)   The simple trend regression is, but not the log-linear trend regression.

B)   No, neither is significant.

C)   The simple trend regression is not, but the log-linear trend regression is.

D)   Yes, both are significant.

The correct answer was D)

The respective t-statistics are 6.7400 / 0.6803 = 9.9074 and 0.1371 / 0.0140 = 9.7929. For 10 degrees of freedom, the critical t-value for a two-tailed test at a five percent level of significance is 2.228, so both slope coefficients are statistically significant.