DETERMINISTIC TREND DETERMINISTIC SEASON MODEL Professor Thomas B Fomby Department of Economics Southern Methodist University Dallas TX June 2008 I Introduction The Deterministic Trend Deterministic Season DTDS model is one of the first time series models proposed to handle trends and seasonality in economic and business data It popularity depended on the relative ease of estimating such a model by the method of ordinary least squares and the interpretability of the model Interestingly this model is a special case of the Unobservable Components model UCM with fixed level 2 0 and fixed slope 2 0 resulting in the deterministic trend specification t 0 0 t fixed dummy seasonals 2 0 assuming an autoregressive factor rt rt 1 t and no irregular component 2 0 The fact that the UCM encompasses the current simple model we are going to entertain just goes to highlight how far time series modeling has come over the last several decades Actually as you will see below we generalize the autoregressive term to be an AR r process as compared to the AR 1 term assumed in the UCM But the point is that essentially the current model is a special case of the UCM II Notation for the DTDS Model Assume the times series y t is observed monthly At first blush one might think naively of writing the DTDS model as y t t 1 Dt1 2 Dt 2 3 Dt 3 12 Dt 12 t 1 1 t 1 t 1 2 t 2 r t r at 2 where y t is the target variable t 1 2 T Dtj is a seasonal dummy variable that takes the value of one if the t th observation is observed in the j th month and 0 otherwise and the errors t follow an AR r process and the errors a t are white noise However a closer look indicates a redundancy in the sample design implied by the parametrization 1 and 2 In fact the full set of seasonal dummy variables Dt1 Dt 2 Dt 12 is pefectly collinear with the intercept To avoid the so called Dummy Variable trap we may take one of three tacts We can drop one of seasonal dummies say the January seasonal Dt1 This implies that the intercept is the January intercept while the coefficients of the other months are increments to the January intercept Let us call this the relative to January parametrization That is the February intercept is 2 the March intercept is 3 etc Of course the dropping of the January dummy is arbitrary and one could just as easily drop any other month For now we will stick with dropping the January dummy and stick with the label chosen for the parametrization We could drop the intercept in our model and then the seasonal dummy coefficients would represent the respective intercepts of the months This of course requires that the regression be run through the origin and in this case some of the classical measures of goodness of fit like the coefficient of determination and the overall F statistic are no longer applicable although traditional hypothesis testing can still be done in this context One might call this the each season has its own intercept parametrization We can keep all of the coefficients in 1 above but impose a restriction on the seasonal dummy coefficients that avoids the Dummy Variable trap One restriction that is often imposed is setting the sum of the seasonal coefficients equal to zero The advantage of this parametrization is that the signs of the coefficients reveal the stronger months those with positive coefficients versus the weaker months those with negative coefficients One might call this the zero sum constraint parametrization 2 As it turns out all of these parametrizations are equivalent to each other in the sense that the coefficients estimates one might get from using one parametrization can easily be translated into the coefficients estimates produced by either of the other two parametrizations Since the relative to January parametrization is easier to implement in SAS especially when using Proc Autoreg to estimate the autocorrelation structure 2 we will pursue this parametrization exclusively in the following discussion and write it as y t t 2 Dt 2 3 Dt 3 12 Dt 12 t 3 t 1 t 1 2 t 2 r t r at 4 In terms of conventional additive time series decomposition the trend is represented by the t part of the model is the y t intercept and is the slope of the deterministic trend line t the part 2 Dt 2 3 Dt 3 12 Dt 12 is seasonal part of the model and t contains the irregular part a t plus the cyclical part 1 t 1 2 t 2 r t r of the model III Examining Some of the Details of the DTDS model Let us look more closely at the DTDS model 3 and 4 In this form the model is a linear trend model as compared to a quadratic trend model If 0 and 0 then generally speaking the y t data is positive at time t 0 and has a positive slope to it Of course we could have 0 which would imply that the data is declining In actuality if the data has curvature in it we could instead model the trend as t t 2 Of course we can make the choice between the linear and quadratic trend forms of the data by closely inspecting the data and as we will later see using tests of hypotheses concerning For now we will assure that the trend in the data we are analyzing is linear hence we assume trend t for now 3 What are the meanings of the seasonal dummy variables Given the specification 1 we see that January s trend line is given by t February s trend line is given by 2 t and December s trend line is given by 12 t Thus the base trend line is January s trend line while the j j 2 3 12 denote the incremental intercept coefficients that distinguishes the other months trend lines in particular the intercepts from the trend line for the January months of each year One can now see why this parametrization of the model is called the Relative to January parametrization Therefore in identifying the seasonal effects both relative and absolute we can to compare the magnitudes of the January intercept 2 the February intercept etc If the j coefficient is positive then the j th month is stronger than January otherwise it is weaker than January Strength of seasonal effect then is relative to January Obviously if all of the j s is positive then by default January s seasonal effect is the weakest The average seasonal effect is of course 12 12 j 2 j 2 2 12 12 12 j 12 j 12 5 Then the strong months in terms of seasonal effect are those whose trend intercept for January and j for the other months are greater than while the weak months have or j intercepts that are less than If one wants to standardize the …