Sample Autocorrelation Functions of Dally Hog Perce Data

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[pic 1]

Samp l e autocorrelation functions of dally hog perce data[pic 2]

The series consists of 250 daily data points covering all the trading days in. 1965. The price variable is the average price in dollars per hundredweight of all hogs sold in the eight regional markets in the United States on a particular day. The sample autocorrelation functions for the original price series and for the first difference of the series are shown in Fig. 15.13.

Observc that the original series is clearly nonstationary. The autocorrelalion function barely declines. even aftCï a 16-period lag. The series is. however. first-order homogeneous. since its first difference is clearly stationary.

In fact. not only is the first-differenced series stationary, but it appears to resemble white noise. since the sample autocorrelation function is close to zero for all k > O. To dctcrminc whether the differenced series is indeed white noise. Ict us calculate the Q statistic for thc first 1 5 lags. The value of this statistic is 14.62. which, with 15 degrees of freedom. is insignificant at the 10 percent Icvcl Wc can thcrcforc conclude that the differenced scrics is white noisc and that the original price series can best bc modeled as d random walk[pic 3]

        [pic 4]        (15.33)[pic 5]

As is the case of most stock markct priccs. our best forecast of PI is its most recent value. and (sadly) thcrc is no model that can help us outperform the market.

        CHAPTER         OF STOQ4.ASnC

152.3 Seasonality and the Autocorrelation Function

We have Just seen that the autxorrelation function can reveal information about the stationarity of a time series. In the remaining chapters of this book we will sec that other information about a time series can obtained from its autocorrelation function. However. we continue here by examining the relationship between the autocorrelation function and the seasonality of a time series.

As discussed Ln the previous chapter. seasonality is just a cyclical behavior that occurs on a regular calendar basis. An example of a highly seasonal time series would be toy sales, which exhlblt a strong peak every Christmas. Sales of ice cream and Iced-tea mix show seasonal peaks each summer In response to Increased demand brought about by warmer weather; Peruvian anchovy production shows seasonal troughs once every 7 years In rcsponse to decreased supply brought about by cyclical changes in the ocean currents.

Often seasonal peaks and troughs arc easy to spot by direct observation of the time series. However. If the time series fluctuates considerably, seasonal peaks and troughs might not be dlsdngulshable from the other fluctuations. Recogni• don of seasonality is Important because it provides information about • •regular. lty" In the series that can aid us In making a forecast. Fortunately. that recognidon can be made easier with the help of the autocorrelation function.

If a monthly time series yt exhibits annual seasonality. the data points in the series should show some degree of correlation with the corresponding data points which lead or lag by 12 months. In other words, we would expect to see some degree of correlation between y, and yt-t,. Since y, and will be correlatcd. as will yr-n and yt-24. we should aiso see correlation between y, and y,-24. Slrnllarly there will be correlation between y, and yr-}4, y, and yt-„, etc. These correladons should manifest themselves In the sample autocorrelation function [pic 6]which will exhlblt peaks at k 12. 24. 36. 48. etc. Thus we can Identify seasonallty by observing regular peaks In the autocorrelation function, even if seasonal peaks cannot be discerned In the time series Itself.[pic 7][pic 8]

[pic 9]

Example 15.3 Hog Productlon As an example, look at the time series for the monthly production of hogs In the United States, shown fn Fig. I S. 14. rt would take a somewhat experienced eye to easily dlscem seasonality In that , series. The seasonality of the series, however. Is readily. apparent In Its sample autocorrelation function. wh!ch Is shown In FIS. 2 S. IS. Note the peaks that occur at k 12, 24, and 36, Indicadng annual cycles In the series.

A aude method of removing the annual qcles ("deseasonailzing" the data) would be to take a 12-month difference, obtaining a' new series Z, — yt — yt-t2. As can be seen In Ffg. 15.16, the sample autocorrelation function for this 12-month dlffere: zed series does not exhibit strong seasonallty. We will see In later chapters that z, represents an extremely simple time-

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[pic 10]

FIGURE 15.14 [pic 11] production (in thousands of hogs per month). Time bounds: January 1962 to December 1971[pic 12]

series model for hog production. since it accounts only for the annual cycle. We can comolete this example by observing that the autocorrelation function in Fig. 15.16 declines only slowly. so tnat there is some doubt as to whether [pic 13]is a stationary series. We therefore first-differenced this series, to obtain w, =

[pic 14]= A(yt — y, -12). The sample autocorrelation function of this series, shown in Fig. 15.17, declines rapidly and remains small, so that we can be confident that w, is a stationary, nonseasonal time series.

FIGURE 15.15

Sample autocorrelation function for hog production series

[pic 15]

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        Ct-LAPTER         Of STOa-tAS•nC

[pic 16]

FIGURE 15.16

Hog production: sample autocorrelatlon tuocUon ot yt — [pic 17][pic 18][pic 19]

[pic 20]

FIGURE 15.17

        Hog pcoauction: sample autocorrelation function ot         — yt-„).

[pic 21]

15.3 TESTING FOR RANDOM WALKS

Do [pic 22]Yœ,abtes such as GNP. employment. and Interest rates tend to reven back to some long-nin trend following a shock. or do they follow random walks? This question Is Important for two reasons. If these variables follow random walks. a regression of one against another can lead to spurious results. (The Gauss-Markov theorem would not hold, for example, because a random walk not have a finite variance. Hence ordinary least squares (OLS) would not yield a consistent parameter estimator.) Detrendlng the variables before running the regression will not help; the detrended series will still be nonsta-