Lecture Notes 3: Trend and SeasonalityECMT 475: Economic Forecasting (Spring 2011)Guangyi Ma1 Modeling and Forecasting Trend1.1 What is trend? slow, long-run evolution in the variable might be the result of slowly evolving preferences, technologies, insti-tutions, and demographics deterministic or stochastic trend linear vs nonlinear trend examples– linear trend: labor force participation rate (female, male)– nonlinear trend: NYSE volume1.2 A model with trend componentWe can specify a model for a univariate time series by using a linear trend,yt= 0+ 1t + utwhere ut i:i:d:N0; 2.To capture the nonlinearity observed in data, we can use a model witha quadratic trend,yt= 0+ 1t + 2t2+ utwhere ut i:i:d:N0; 2.1Sometimes, othe r types of nonlinear trend are more appropriate, such asexponential trend (log-linear trend), why?yt= 0exp (1t + ut)ln (yt) = ln (0) + 1t + ut1.3 Estimating trend modelsWe use least square estimation again, since these models are special casesof linear regression models. Note that in the "quadratic trend" model, ytis not a linear function of t, but it is a linear function oft; t2. Similarly,the exponential trend model can be transfered into a linear regression modelafter taking log on yt.1.4 Forecasting trendOnce we get an estimate from the regression model, we can make out-of-sample forecast: point, interval, or even density forecast if needed. All thedetails are described in previous reviews.1.5 Issues of model selectionOne important question is, how do we pick a special model among severalcompeting ones to …t a series of observations? More generally, when we runa linear regression, which set of X variables should we include in the righthand side? Here our goal is to …nd a model with the smallest out- of-sam ple1-step-ahead mean squared prediction error. Several critera follow. MSE or R2:MSE =1TXTt=1e2tR2= 1 PTt=1e2tPTt=1(yt y)2They are equivalent since smaller MSE implies larger R2. This is not agood model selection criterion because it does not penalize the degree2of freedom. One cannot choose the …tted model corresponding to thelargest R2since R2will keep increasing (at least not decrease) whenmore variables included. A mode…ed MSE or R2:s2=1T  kXTt=1e2t=TT  kMSER2= 1 PTt=1e2t= (T  k)PTt=1(yt y)2= (T  1)They are equivalent too! This selection criterion does penalize thedegree of freed om. AIC or SIC:AIC = exp2kT1TXTt=1e2t= exp2kTMSESIC = TkT1TXTt=1e2t= TkTMSEThey are di¤erent, both penalize the degree of freedom highter thanthe mode…ed MSE. We usually need both of them as reference formodel selection.2 Modeling and Forecasting Seasonality Nature and sources of seasonality– Seasonality comes from calendar cycles due to technology (mostlyweather related), pref erenc e, institutions, etc.– Examples: gasoline sales, liquor sales, durable goods sales, etc. Modeling seasonality: dummy variables! Extension: more general calendar e¤ects Examples: holiday variation, trading-day variation, Monday e¤ect3 Forecasting seasonality: all regular tools