GDP Advance Estimate, 2016Q4

Transcription

1 GDP Advance Esimae, 26Q4 Friday, Jan 27 Real gross domesic produc (GDP) increased a an annual rae of.9 percen in he fourh quarer of 26. The deceleraion in real GDP in he fourh quarer refleced a downurn in expors, an acceleraion in impors, a deceleraion in PCE, and a downurn in federal governmen spending Real GDP increased.6% in 26

2 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

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

4 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

5 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

6 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

7 Example Revisied: PCE U.S. Real Personal Consumpion Expendiure From Naional Income Accouns Quarerly, percenage change from previous period, annualized 947q2 o 26q4 use realgdpgrowh regress pce predic yp predic s, sdf gen yp = yp.645*s gen gp2 = yp +.645*s sline pce yp yp yp2, lpaern (solid solid dash dash)

8 In-Sample Predicion Inervals PCE, fied values, and inervals q 96q 97q 98q 99q 2q 2q 22q ime Real Personal Consumpion Expendiures yp Fied values yp2

9 Ou of Sample sappend, add(2) predic p if ime>q(26q4) predic s, sdf gen p = p.645*s gen p2 = p +.645*s sline pce yp p p p2 if ime>q(2q4), lpaern (solid solid dash shordash shordash) q 25q 2q 25q 22q ime Real Personal Consumpion Expendiures poin forecas p2 Fied values p

10 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

11 Sae and Local Governmen Spending Percenage Growh Rae (Quarerly) Average for : 3.% Bu his has no been ypical in recen years q 96q 97q 98q 99q 2q 2q 22q ime Real Governmen Consumpion Expendiures and Gross Invesmen: Fied Sae valuesand Local

12 Alernaives Subsample esimaion Esimae he mean on subsamples Forecass are based on he mos recen Dummy Variable formulaion ( y Ω ) + β 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 = β E + h d =

13 Forecas Linear Regression y +h on d Example: Sae and Local Governmen Percenage Growh Mean breaks in 97q and 22q generae d=(ime>=q(97q)) generae d2=(ime>=q(22q)) regress sae d d2 predic db sline sae sb Source SS df MS Number of obs = 279 F(2, 276) = 4.89 Model Prob > F =. Residual R-squared =.2286 Adj R-squared =.223 Toal Roo MSE = sae Coef. Sd. Err. P> [95% Conf. Inerval] d d _cons

14 Fied Ou-of-sample forecas falls from 3.2% o %! q 96q 97q 98q 99q 2q 2q 22q ime Real Governmen Consumpion Expendiures and Gross Invesmen: Fied Sae valuesand Local

15 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.

16 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

17 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 )

18 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.

19 Example Labor Force Paricipaion Rae From BLS Monhly, , Seasonally adjused Men and Women, ages 2+ Percenage of populaion in labor force (employed plus unemployed divided by populaion) We will esimae on Forecas 99-26

20 Example freduse LNS325 LNS326 rename LNS325 men rename LNS326 women smkim ime, sar(948m) sse ime sline women if ime<m(99m) sline men if ime<m(99m)

21 Women s Labor Paricipaion Rae Labor Force Paricipaion Rae - 2 years and over, Women m 96m 97m 98m 99m ime

22 Men s Labor Paricipaion Rae Labor Force Paricipaion Rae - 2 years and over, Men m 96m 97m 98m 99m ime

23 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 = β + β Time T In his model, β is he expeced period-operiod change in he rend T

24 Quadraic Trends An alernaive model is a quadraic rend T = β + β Time + β Time 2 2

25 Example 2 Transacion Volume, S&P Index From Yahoo Finance GSPC, closing price and average volume Weekly, 95-curren Saa file s&p.da on course webpage We esimae on Forecas 994-curren

26 Transacion Volume volume.e+8 2.e+8 3.e+8 4.e+8 5.e+8 jan95 jan96 jan97 jan98 jan99 ime index

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) lvolume jan95 jan96 jan97 jan98 jan99 ime index 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 3 Real GDP We will esimae on , forecas Also use an exponenial rend

32 Real GDP Real Gross Domesic Produc q 96q 97q 98q 99q 2q 2q 22q ime

33 Ln(Real GDP) ln_rgdp q 96q 97q 98q 99q 2q 2q 22q ime

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 Assignmens Read Diebold hrough Chapers Problem Se # 3 Due Tuesday (2/7) Read Chaper 2 from The Signal and he Noise Reading Reflecion Due Thursday (2/2)