Seasonal Dummy Model

1 Seasonal Dummy Model Deterministic seasonality Stcan be written as a function of Seasonal Dummy variables Let s be the Seasonal frequency s=4 for quarterly s=12 for monthly Let D1t, D2t, D3t,.., Dstbe Seasonal dummies D1t= 1 if s is the first period, otherwise D1t= 0 D2t= 1 if s is the second period, otherwise D2t= 0 At any time period t, one of the Seasonal dummies D1t, D2t, D3t,.., Dstwill equal 1, all the others will equal Dummy Model Deterministic seasonalitya linear function of the Dummy variablesitsiitDDecembertifFebruarytifJa nuarytifS == ====11221 MMEstimation Least squares regression You can either Regress yon all the Seasonal dummies, omitting the intercept, or Regress yon an intercept and the Seasonal dummies, omitting one Dummy (one season, December) You cannot regress on both the intercept plus all Seasonal dummies, for they would be collinear and ++=+= ==+111 Interpreting Coefficients In the modelthe intercept = sis the seasonality in the omitted season.

2 The coefficients i= i s are the difference in the Seasonal component from the s th =+=11 STATA Programming If the time index is tand is formatted as a time index, you can determine the period using the commandsgenerate m=month(dofm(t))generate q=quarter(dofq(t))for monthly and quarterly data , respectively(See dates and times in STATA data manual)Creating Dummies If mis the month (1 for January, 2 for February, etc.), then generate m1=(m==1) This creates a Dummy variable m1 for January Then regress y m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11or regress y m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12, noconstant Easier Type in the regressor list regress y This includes dummies for months 1 through 11, omits 12 Same as mechanically listing the eleven dummies, but easier.

3 It is important that m be the numerical month (1 for January, 2 for February, etc.)EstimationEstimated Seasonality Housing StartsJanuary91 February95 March127 April144 May150 June148 July142 August140 September132 October137 November114 December96 Housing Starts, by year, and estimated seasonalityPredicted ValuesExample 1 Unemployment RateUnemployment Rate, by year, and estimated seasonalityPredicted ValuesExample 2 Gasoline SalesGasoline Sales, by year, and estimated seasonalityPredicted ValuesApplication Weekly data Unemployment Insurance Claims Department of Labor Issued Weekly Important indicator for unemploymentUnemployment ClaimsNot Seasonally AdjustedUnemployment ClaimsOfficial Seasonally Adjusted SeriesEstimationEstimated Seasonal ProcessSeasonally Adjusted (by Dummy Variable Method)

4 Other types of seasonality Daily data Day of the week Handle by including Dummy variables for each day High frequency data Include hourly or time of day indicators Holiday effects Flower sales big on Valentines Day, Mothers Day, Easter, yet these days can move around Trading day/business day variation Number of trading days/business days varies across months Can divide by number of trading days, or include as a regressorSeasonal + Cycle Consider a components Model with Seasonal and AR(1) cycle The Seasonal Stis a set of Seasonal dummiestttttteCCCSy+=+= 1 itsiitDS ==1 Transformation Take the first equation and lag it once Multiply it by Then subtract it from the first equation to findtttttteCCCSy+=+= 1 111 +=tttCSy111 +=tttCSy ttttteSSyy+ += 11 Seasonal Representation We find When the Seasonal Stis a set of Seasonal dummies, one for each season, this equation suggests a regression on yt 1 Seasonal dummies Lagged Seasonal dummiesttttteSSyy+ += 11 Redundant But lagged Seasonal dummies are redundant with the original Seasonal dummies The set of lagged Dummy variables are collinear with the current Dummy variables Given that you know this month is February.

5 There is no information in knowing that last month was January. The lagged dummies can be (should be) omittedSeasonal + Cycle We have found that the regression Model isorttitsiiteyDy++= = 11 ttitsiiteyDy+++= = 1110 AR(p) Case If the cycle is an AR(p) we have Estimate by least squares Linear ForecastingtptptitsiiteyyDy++++= = L111 Trend+ Seasonal +Cycle A full Model istptpttitsiitttttteCCCDStTCSTy+++==+=++ = = L11121 Regression Model The implied regression Model is This can be estimated by least squares It is a complete forecasting modeltptptitsiiteyytDy+++++= = L111 Example: NSA Unemployment RateRegress on Dummies plus AR(12)AR CoefficientsFitted ValuesLast 2 years12 month forecastForecasting with Seasonal Dummy To forecast in STATA with Seasonal dummies, the Dummy variables must be defined for the forecast period Afteryou use the tsappendcommand, you create the month m=month(dofm(t)) m=month(dofm(t)) Otherwise m will have missing values for the forecast periodExample: Retail Sales Census Bureau Monthly Retail Sales Not Seasonally Adjusted and Seasonally Adjusted Sales listed by variety of categories 1992 2009 From Census Bureau SpreadsheetJan.

6 2009 Feb. 2009 Mar. 2009 Apr. 2009 May 2009 Jun. 2009 NOT ADJUSTEDR etail and food services sales, total313,593304,056334,149336,155354,668 351,418 Retail sales and food services excl motor vehicle and parts262,237252,560275,000277,938294,559 289,242 Retail sales, total277,402269,015295,520298,119313,979 312,547 Retail sales, total (excl. motor vehicle and parts dealers)226,046217,519236,371239,902253, 870250,371 GAFO(1)83,32382,91887,47787,24893,06988, 330 Motor vehicle and parts dealers51,35651,49659,14958,21760,10962, 176 Automobile and other motor vehicle dealers45,38545,54452,49851,55853,52855, 239 Automobile dealers41,79041,63246,83445,37346,60148, 405 New car dealers36,03235,30440,57539,49240,94142, 530 Used car dealers5,7586,3286,2595,8815,6605,875 Automotive parts, acc.

7 , and tire stores5,9715,9526,6516,6596,5816,937 Furniture, home furn, electronics, and appliance stores16,06915,70015,53014,54515,36215,5 19 Furniture and home furnishings stores7,4287,2197,6017,2937,6897,728 Furniture stores4,2304,2964,2623,9694,2614,148 Home furnishings stores3,1982,9233,3393,3243,4283,580 Floor covering stores1,3811,3351,4481,4751,4621,673 All other home furnishings stores1,6981,4801,7621,7251,8511,790 Electronics and appliance stores8,6418,4817,9297,2527,6737,791 Appl.,TV, and other elect. stores6,7066,6155,9365,3745,8275,857 Household appliance stores1,3061,2011,2491,2521,3171,349 Radio, , and other elect. stores5,4005,4144,6874,1224,5104,508 Computer and software stores1,7341,7221,7901,6791,6121,706 Liquor Sales Beer, wine and liquor Sample: 1992 2009 Not Seasonally Adjusted Big spike in DecemberLiquor Sales (Millions of $)ln(Liquor Sales)Residual from Linear TrendSeasonal DummyResiduals after Seasonal DummiesFull EstimationAR CoefficientsFitted ValuesLast 3 yearsResiduals12 Month Forecast12 month forecast The big jump down in the forecast for January 2010 is because of the Seasonal Dummy effect