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Time Series Analysis Materials for this lecture Lecture 5 Lags.XLS Lecture 5 Stationarity.XLS Read Chapter 15 pages 30-37 Read Chapter 16 Section 15

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Time Series Analysis Forecast a variable based only on its past values We are going to focus on the application and less on the estimation calculations Simetar estimates TS models easily

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Time Series Analysis A comprehensive forecasting methodology for univariate distributions Best for variables unrelated to other variables in a structural model Autoregressive process will prevail Ŷ t = f(Y t-1, Y t-2, Y t-3, Y t-4, … ) Assumptions are –E(Mean of ê) = 0.0, i.e., data are stationary –Variance of ê t constant

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Time Series Analysis Steps for estimating a times series model are: –Graph the data series to see what patterns are present – trend, cycle, seasonal, and random –Test data for Stationarity with a Dickie Fuller Test If original series is not stationary then difference it until it is Number of Differences (p) to make a series stationary is determined using the D-F Test Use the stationary (differenced) data series to determine the number of Lags that best forecasts the historical period Given a stationary series, use the Schwarz Criteria (SIC) or autocorrelation table to determine the best number of lags (q) to include when estimating the model –Estimate the AR(p,q) Model and make a forecast using the Chain Rule

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Time Series Analysis Assumptions Assume series you are forecasting is random –No trend, seasonal, or cyclical pattern remains in series –Series is stationary or “white noise” –To guarantee this assumption we make the data stationary Assume the variability is the same for the future as for the past –σ 2 T+i = σ 2 t –Can test for constant variance by testing correlation over time –Crucial assumption because if σ 2 changes overtime, then forecast will explode overtime

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Make Data Series Stationary Take differences of the data to make it stationary First difference of Y to calculate D 1,t D 1,t = Y t – Y t-1 Calculate first difference of D 1,t for a Second Difference of Y or D 2,t = D 1,t – D 1,t-1 Calculate first difference of D 2,t for a Third difference of Y or D 3,t = D 2,t – D 2,t-1 Stop differencing data when series is stationary

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Make Data Series Stationary Example Difference table for a time series data set –D 1t in period 1 is 0.41= 71.47 – 71.06 –D 1t in period 2 is -1.41= 70.06 – 71.47 –D 2t in period 1 is -1.82 = -1.41 – 0.41 –D 2t in period 2 is 1.66 = 0.25 – (– 1.41) –Etc.

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Test for Stationarity Dickie-Fuller Test for stationarity –First D-F test -- is original data stationary? –Estimate regression Ŷ = a + b D 1,t –Calculate the t-ratio for b to see if a significant slope exists –D-F Test statistic is the t-statistic; if more negative than -2.9, we declare the dependent series (Ŷ in this case) stationary at alpha 0.05 level (it’s a 1 tailed t test) –Thus a D-F statistic of -3.0 is more negative than -2.90 so the original series is stationary and differencing is not necessary

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Test for Stationarity Second D-F Test for stationarity of the D 1,t series, in other words, we are testing if the Y series be stationary with one differencing? –Estimate regression for D 1,t = a + b D 2,t –t-statistic on slope b is the second D-F test statistic

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Test for Stationarity Third D-F Test for stationarity of the D 2,t series, in other words will the Y series be stationary with two differences? –Estimate regression for D 2,t = a + b D 3,t –t-statistic on slope b is the third D-F test statistic Continue with a fourth and fifth DF test if necessary

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Test for Stationarity How does D-F test work? –A series is stationary, if it oscillates about the mean and has a slope of zero –If a 1 period lag Y t-1 is used to predict Y t, they are good predictors of each other because they have the same mean. The t-statistic on b will be significant for D 1,t = a + b D 2,t –Lagging the data 1 period causes the two series to be inversely related so the slope is negative in the OLS equation. That explains why the “t” coefficient of -2.9 is negative. The -2.9 represents a 5% 1 tail test statistic.

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Test for Stationarity Estimated regression for: Ŷ = a + b D1,t DF is -1.868 the t ratio See a trend in original data and no trend in the residuals Intercept not zero so mean of Y is not constant

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Test for Stationarity Estimated regression for: D1,t = a + b D2,t DF is -12.948 the t ratio See the residuals oscillate about a mean of zero, no trend in either series Intercept is 0.121 or about zero, so the mean is more likely to be constant

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Test for Stationarity Estimated regression for: D2,t = a + b D3,t DF is -24.967 the t ratio Intercept is about zero

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Test for Stationarity Dickie Fuller test in Simetar =DF ( Data Series, Trend, No Lags, No. Diff to Test) Where: Data series is the location of the data, Trend is True or False No. Lags is optional if data are to be lagged No. Diff is the number of differences to test

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Test for Stationarity Number of differences that make the data series stationary is the one that has a DF test statistic more negative than -2.9 Here it is 1 difference no matter if trend included or not Lecture 5

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Summarize Stationarity Y t is the original data series D i,t is the i th difference of the Y t series We difference the data to make it stationary –To guarantee the assumption that the mean is constant Dickie Fuller test is used to determine the D i,t no. of differences needed to make series stationary =DF(Data, Trend,, No. of Differences) Test 0 to 6 differences with and without trend and zero lags using =DF() Select the difference with a DF test stat. More negative than -2.90

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NEXT We Determine Number of Lags Once we have determined the number of differences we know what form the auto regressive (AR) model will be fit –AR ( No. Differences, No. Lags) or AR(p,q) If the series is stationary with 1 difference we will fit the model D 1,t = a + b 1 D 1,t-1 + b 2 D 1,t-2 + … The only question that remains is how many lags (q) of D i,t will we need to forecast the series –To determine the number of lags we use several tests

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Number of Lags for TS Model Partial autocorrelation coefficients used to estimate number of lags for D i,t in model. Assume D 1,t is stationary then: To test for one lag use b 1 from regression model D 1,t = a + b 1 D 1,t-1 To test for two lags use b 2 from regression model D 1,t = a + b 1 D 1,t-1 + b 2 D 1,t-2 To test for three lags use b 3 from regression model D 1,t = a + b 1 D 1,t-1 + b 2 D 1,t-2 + b 3 D 1,t-3 After each regression we only use the beta (b i ) for the longest number of lags, i.e., the bold ones above Use the t stat on the b i to determine contribution of the lags to explain D 1,t So this calls for running lots of regressions with different numbers of lags, right?

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Number of Lags for TS Model Find the optimal number of lags the easy way Use Simetar to build a Sample Autocorrelation Table (SAC) =AUTOCORR(Data Series, No. Lags, No. Diff) Pick best no. of lags based on the last lag with a statistically significant t value

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Number of Lags for TS Model Bar chart of autocorrelation coefficients in Sample AUTOCORR() Table The explanatory power of the distant lags is not large enough to warrant including in the model, based on their t stats, so do not include them.

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Autocorrelation Charts of Sample Autocorrelation Coefficients (SAC) 1 1 Lag k A SAC that cuts off Dying down in a dampened exponential fashion – no oscillation Dying down in a dampened exponential fashion – oscillation 1 1 Lag k 1 Lag k Dying down quickly Dying down extremely slowly

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Note: Partial vs. Sample Autocorrelation Partial autocorrelation coefficients (PAC) show the contribution of adding one more lag. –It takes into consideration the impacts of lower order lags –A beta for the 3rd lag shows the contribution of 3rd lag after having lags 1-2 in place D 1,t = a + b 1 D 1,t-1 + b 2 D 1,t-2 + b 3 D 1,t-3 Sample autocorrelation coefficients (SAC) show contribution of adding a particular lag. –A SAC for 3 lags shows the contribution of just the 3 rd lag. Thus the SAC does not equal the PAC

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Number of Lags for Time Series Model Some authors suggest using SAC to determine the number of differences to achieve stationarity –If the SAC cuts off or dies down rapidly it is an indicator that the series is stationary –If the SAC dies down very slowly, the series is not stationary This is a good check of the DF test, but we will rely on the DF test for stationarity

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Number of Lags for Time Series Model Schwarz Criteria (SIC) tests for the optimal number of lags in a TS model Goal is to find the number of lags which minimizes the SIC In Simetar use the ARLAG() function which returns the optimal number of lags based on SIC test =ARLAG(Data Series, Constant, No. of Differences)

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Schwarz Criteria and ARLAG Tables Pick the no. of lags to Minimize the SC –> 1 lag ARLAG indicates no. of lags to minimize SC –> 1 lag, given the no. of differences. Autocorr finds no. of lags where AC drops –> 1 lag

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Summarize Stationarity/Lag Determination Make the data series stationary by differencing the data –Use the Dickie-Fuller Test (df < -2.90) to find D i series that is stationary Use the =DF() function in Simetar Determine the number of Lags for the D i series –Use the sample autocorrelation coefficients for alternative lag models =AUTOCORR() function in Simetar –Minimize the Schwarz Criteria =ARLAG() or =ARSCHWARZ() functions in Simetar

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