Estimation Uncertainty

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1 Esimaion Uncerainy The sample mean is an esimae of β = E(y +h ) The esimaion error is = + = T h y T b ( ) = = + = + = = = T T h T h e T y T y T b β β β

2 Esimaion Variance Under classical condiions, where σ 2 =var(e ) var ( b ) 2 σ = T The sandard error for b is an esimae of he sandard deviaion sd ( b ) = σˆ T 2

3 Forecas Variance When he sample mean b is used as he forecas for y T+h hen he predicion error is T + h b = et + h + β b which is he sum of he forecas error e T+h and he esimaion uncerainy β b. The forecas variance is y var ( y b ) = var( e ) + var( β b ) T + h T + h 2 2 σ = σ + T = 2 + σ T

4 Sandard Deviaion of Forecas The sandard deviaion of he forecas is he esimae s T + h = 2 + ˆ σ T This is slighly larger han he regression sandard deviaion σˆ Calculaed in STATA afer a regression using he sdf opion o he predic command: predic s, sdf This creaes variable s

5 Normal Forecas Inervals Le ŷ T+h be a forecas for y T+h The predicion error is y T+h ŷ T+h Le s T+h be he s. deviaion of he forecas If he predicion errors are normally disribued, he ( α)% forecas inerval endpoins are where z α/2 and z α/2 are he α/2 and α /2 quaniles of he normal disribuion e.g. ŷ T+h ±.64 s T+h for a 9% inerval 2 / 2 / ˆ ˆ α α = + = z s y U z s y L h T h T h T h T h T h T

6

7 Ou of Sample

8 Ou of Sample

9 Mean Shifs Someimes he mean of a series changes over ime I can drif slowly, or change quickly Possibly due o a policy change In his case, forecasing based on a consan mean model can be misleading

10 Sae and Local Governmen Spending Percenage Growh Rae (Quarerly) Average for : 3.24% Bu his has no been ypical in recen years.

11 Alernaives Subsample esimaion Esimae he mean on subsamples Forecass are based on he mos recen Dummy Variable formulaion ( y Ω ) E + h = β + βd d = ( τ ) τ is he breakdae The dae when he mean shifs The coefficien β is he mean before =τ The coefficien β is he shif a =τ The sum β +β is he mean afer =τ

12 Forecas Linear Regression y +h on d Example Sae and Local Governmen Percenage Growh Mean breaks in 97q and 22q

13 Fied Ou of sample forecas falls from 3.2% o.4%!

14 Should you use Mean Shifs? Only afer grea hesiaion and consideraion. Should use shifs and breaks relucanly and wih care. Do you have a model or explanaion? Wha is he forecasing power of a mean shif? If hey have happened in he pas, will here be more in he fuure? Ye, if here has been an obvious shif, a simple consan mean model will forecas erribly.

15 How o Selec Breakdaes Judgmenal Daes of known policy shifs Imporan evens Economic crises Informal daa based Visual inspecion Formal daa based Esimae regression for many possible breakdaes Selec one which minimizes sum of squared error This is he leas squares breakdae esimaor

16 Trend Models A rend model is T = g ( Time where Time is he ime index. In STATA, Time is an ineger sequence, normalized o be zero a firs observaion of 96. Mos common models Linear Trend Exponenial Trend Quadraic Trend Trends wih Changing SLope )

17 Warning: Be skepical of Trend Models While in some cases, rend forecasing can be useful. In many cases, i can be hazardous. We will examine some examples from anoher exbook (Diebold: Elemens of Forecasing) They did no forecas well ou of sample. A consrucive alernaive is o forecas growh raes, as we did for consumpion expendiure.

18 Example Labor Force Paricipaion Rae From BLS Monhly, , Seasonally adjused Men and Women, ages 25+ Percenage of populaion in labor force (employed plus unemployed divided by populaion) We will esimae on Forecas

19 Women s Labor Paricipaion Rae

20 Men s Labor Paricipaion Rae

21 Linear Trend Model The labor force paricipaion raes have been smoohly and linearly increasing (for women) and smoohly and linearly decreasing (for men) over This suggess a linear rend T = β + β Time In his model, β is he expeced period operiod change in he rend T

22 Example 2 Reail Sales, Curren Dollars From Census Bureau Monhly, 955 2, seasonally adjused This paricular series disconinued afer 2 We will Forecas 992 curren

23 Reail Sales

24 Quadraic Trends The reail sales series has been increasing smoohly over , bu no linearly. To model his we will use a quadraic rend T = β + β Time + β Time 2 2

25 Example 3 Transacion Volume, S&P Index From Yahoo Finance Weekly, 95 curren We esimae on Forecas 994 2

26 Transacion Volume

27 Exponenial Trend To model his we will use an exponenial rend β + β T = e Time The exponenial rend is linear afer aking (naural) logarihms ln ( T ) = β + βtime This is ypically esimaed by a linear model afer aking logs of he variable o forecas

28 Ln(Volume) In logarihms, rend is roughly linear.

29 Exponenial Trends Mos economic series which are growing (aggregae oupu, such as GDP, invesmen, consumpion) are exponenially increasing Percenage changes are sable in he long run These series canno be fi by a linear rend We can fi a linear rend o heir (naural) logarihm

30 Linear Models The linear and quadraic rends are boh linear regression models of he form or T = β + β x T = β + βx + β2x2 where x = Time x 2 = Time 2

31 From BEA Quarerly, Example 4 Real GDP We will esimae on , forecas Also use an exponenial rend

32 Real GDP

33 Ln(Real GDP)

34 Linear Forecasing The goal is o forecas fuure observaions given a linear funcion of observables In he case of rend esimaion, hese observables are funcions of he ime index In oher cases, hey will be oher funcions of he daa In he model T = β + βx he forecas for y +h is ŷ +h =b +b x where b and b are esimaes

35 Esimaion How should we selec b and b? The goal is o produce a forecas wih low mean square error (MSE) The bes linear forecas is he linear funcion β +β x ha minimizes he MSE E 2 ( y yˆ ) = E( y β β x ) 2 + h + h + h We do no know he MSE, bu we can esimae i by a sample average

36 Sum of Squared Errors Sample esimae of mean square error is he sum of squared errors S n n n ( β, β ) = ( y β β x ) = + h The bes linear forecas is he linear funcion β +β x ha minimizes he MSE, or expeced sum of squared errors. Our sample esimae of he bes linear forecas is he linear funcion which minimizes he (sample) sum of squared errors. This is called he leas squares esimaor 2

37 Leas Squares The leas squares esimaes (b,b ) are he values which minimize he sum of squared errors n 2 S ( β, β ) = ( y β β x ) n n = + h This produces esimaes of he bes linear predicor he linear funcion β +β x ha minimizes he MSE

38 Muliple Regressors There are muliple regressors For example, he quadraic rend T T = β + βx + β2x2 = β + β Time + β Time 2 2 The bes linear predicor is he linear funcion β +β x +β 2 x 2 ha minimizes he MSE E ( ) 2 y yˆ = E( y β β x β x ) 2 + h + h + h 2 2

39 Muliple Regression The sample esimae of he bes linear predicor are he values (b,b,b 2 ) which minimize he sum of squared errors In STATA, use he regress command ( ) ( ) = + = n h n x x y n S ,, β β β β β β

40 Example Women s Labor Force Paricipaion Rae

41 Regression Esimaion

42 In Sample Fi

43 Residuals Residuals are difference beween daa and fied regression line eˆ = = y y + h + h T b b Time

44 Residual Plo

45 In Sample Fi Compue wih predic command Fi looks good

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