Forecasting. Simultaneous equations bias (Lect 16)

Transcription

1 Forecasting. Simultaneous equations bias (Lect 16) Ragnar Nymoen University of Oslo 11 April / 20

2 References Same as to Lecture 15 (as a background to the forecasting part) HGL, Ch (forecasting with an ARDL model), and Ch (), Ch BN Kap (skip for now) 2 / 20

3 Model based forecasting, an example Medium term macro model C t : private consumption in year t (in constant prices, eg.2010) GDP t, TAX t and I t are gross domestic product, net taxes and investments and gov.exp. a - e are parameters of the macroeconomic model ɛ Ct and ɛ TAXt are independent disturbances with classical properties conditional on I t and C t 1. C t = a + b(gdp t TAX t ) + cc t 1 + ɛ Ct (1) TAX t = d + egdp t + ɛ TAXt (2) GDP t = C t + I t (3) C t, GDP t and TAX t are endogenous, C t 1 is predetermined. Assume that I t is strictly exogenous with E (I t ) = µ I and Var(I t ) = σi 2. For simplicity, we will use I t = µ I + ɛ It (4) where is independent of ɛ Ct and ɛ TAXt, and has classical properties conditional on I t and C t 1. 3 / 20

4 Reduced form Suppose that our purpose is to forecast C T +1, C T +2,...,C T +H based on data up to and including period T. Given some premises that will not be discussed (but in E 4160), the best forecast for C T +1 is the conditional forecast. This implies that we generate forecasts from the reduced form equation for C t from (1)-(3) a + bd b(1 e) c C t = + I t + C t 1 + ɛ Ct (be)ɛ TAXt (1 b(1 e)) (1 b(1 e)) (1 b(1 e)) (1 b(1 e)) }{{}}{{}}{{}}{{} β 0 β 1 β 2 ε t This is an ARDL model! β j (j = 0, 1, 2) are reduced form coefficients that can be estimated from a sample t = 1, 2,..., T. Question: Given the model specification, what are the properties of the OLS estimators ˆβ j (j = 0, 1, 2)? 4 / 20

5 1-step ahead forecast I Since E (ε T +1 C T, I T ) = 0 the model consistent best forecast for T + 1 becomes: E (C T +1 C T, I T ) = β 0 + β 1 E (I T +1 C T, I T ) + β 2 C T (5) Next, from the exogeneity of I t : E (I T +1 C T, I T ) = E (I T +1 ) = µ I so the forecast for C T +1 can be written as E (C T +1 C T, I T ) = β 0 + β 1 µ I + β 2 C T Note that I T +1 has been forecasted by E (I t ) = µ I 5 / 20

6 1-step ahead forecast II Reminds us that it necessary to also forecast the exogenous variable! In practice forecasters often use subjective forecasts for exogenous variables, and often present alternatives scenarios. Here we keep it clean and have used the mathematical expectation consistent with the model assumptions. The practical problem with using E (C T +1 C T, I T ) as a forecast is that the parameters are unknown. In practice we therefore replace E (C T +1 C T, I T ) by the estimated expectation: Ĉ T +1 = ˆβ 0 + ˆβ 1 ˆµ I + ˆβ 2 C T (6) where ˆβ 0, ˆβ 1, ˆβ 2 are OLS estimates of the reduced form ARDL model and ˆµ I = Ī, using data from t = 1, 2,..., T. 6 / 20

7 1-step ahead forecast III Compare the prediction of the out-of-sample Y j in Lecture 5. 7 / 20

8 h-period ahead dynamic forecasts Define (to simplify notation) ˆγ = ˆβ 0 + ˆβ 1 ˆµ I The forecasts for 1,2, 3 and 4 periods ahead from T becomes; Ĉ T +1 = ˆγ + ˆβ 2 C T Generally, for forecast horizon h: Ĉ T +2 = ˆγ(1 + ˆβ 2 ) + ˆβ 2 2C T Ĉ T +3 = ˆγ(1 + ˆβ 2 + ˆβ 2 2) + ˆβ 3 2C T Ĉ T +4 = ˆγ(1 + ˆβ 2 + ˆβ ˆβ 3 2) + ˆβ 4 2C T Ĉ T +h = h 1 ˆβ j 2 ˆγ + ˆβ h 2C T h = 1, 2,..., H (7) j=0 8 / 20

9 Long-horizon forecast I If 1 < ˆβ h 2 < 1 we get from Ĉ T +h = h 1 j=1 ˆβ j 2 ˆγ T + ˆβ h 2C T h = 1, 2,..., H that Ĉ T +h where (from Lecture 15), Ê (C t ) = h ˆγ 1 ˆβ 2 ˆγ 1 ˆβ 2 is the estimated equilibrium value (steady state value) of C t. We have the important result that: As h grows, the dynamic forecasts converge to the equilibrium of the forecasted variable. Dynamic econometric forecasts are equilibrium correcting. 9 / 20

10 Forecast errors: Bias Bias: h 1 C t+h Ĉ T +h = (γ ˆγ) ˆβ h 2 + (β h 2 ˆβ h h 1 2)C T + β j 2 ε t+j j=0 j=0 E ( ) h 1 [ ] C t+h Ĉ T +h = E (γ ˆγ) ˆβ h 2 + C T E (β h 2 ˆβ h 2) h = 1, 2,... H j=0 (8) Cannot prove that E ( C t+h Ĉ T +h ) = 0 for any horizon h. But biases can be small if the estimation sample period is sufficiently large (cf. Lecture 15) 10 / 20

11 Forecast errors: Variance Bias: h 1 C t+h Ĉ T +h = (γ ˆγ) j=0 Var ( ) h 1 C t+h Ĉ T +h = Var [(γ ˆγ) ˆβ h 2 j=0 ˆβ h 2 + (β h 2 ˆβ h h 1 2)C T + ] j=0 β j 2 ε t+j + σ 2 h 1 β 2j 2 h = 1, 2,... H j=0 (9) The first part corresponds to the estimation uncertainty we had in Lecture 5 That part will be small if the sample period is sufficiently large (cf. Lecture 15) This means that the second part of (9) will dominate the variance of the forecast error, and: Var ( C t+h Ĉ T +h ) h Var(C t ) = σ2 1 β 2 2 We expect that the variance of the forecast error converge to the theoretical variance of the forecasted variable. 11 / 20

12 Illustrating the forecasting theory Assume that the parameters of the macro model are as in: C t = (GDP t TAX t ) C t 1 + ɛ Ct (10) TAX t = GDP t + ɛ TAXt (11) I t = ɛ It (12) GDP t = C t + I t but that we use the estimated reduced form ARDL for C t to forecast C t+j. Generate data for Use t = 1, 2,..., 101 to estimate ARDL for C t Forecast C 101+h, h = 1, 2,..., 10 from estimated ARDL Using true structural parameters as in (10)-(13) (13) 12 / 20

13 259 Forecast with estimated reduced form coefficients Forecast with known, true model parmeters C C prediction intervals (95 %) / 20

14 Summary I It works! The prediction intervals contain the actuals for C! We have no forecast failure In particular: Estimation uncertainty is not a main source of forecast failures (actuals outside) Intuitively, this is because C T +h fluctuates around its equilibrium, and Ĉ T +h converges to the same equilibrium Examples from real-life forecasting in class 14 / 20

15 Summary II It works under certain assumptions! The estimated model is correctly specified econometrically No structural break in the forecast period When forecasts fail badly (and they do), it is usually because of structural breaks in the forecast period. In our example, a drop of µ I from 100 to 90 would produce large and persistent forecast errors for C In practice: Important to adapt as quickly as possible after a structural break. 15 / 20

16 Structural parameter Assume now that our purpose is the estimation of the structural parameter b, the marginal propensity to consume b in the above model For exposition, we can simplified and static structural model C t = a + b(gdp t ) + ɛ Ct (14) GDP t = C t + I t (15) I t = µ I + ɛ It (16) Assumptions about structural disturbances: E (ɛ Ct I t ) = 0, Var(ɛ Ct I t ) = σ 2 C (17) E (ɛ It ) = 0, Var(ɛ Ct ) = σ 2 I (18) Kov(ɛ Ct, ɛ It ) = 0 (19) In this model, the OLS estimator for b cannot be consistent. Why? 16 / 20

17 OLS estimator We know that the OLS estimator is ˆb = T t=1(gdp t GDP)C t T ( t=1 GDPt GDP ) 2 = b + T t=1 GDP t (ɛ Ct ɛ C ) T ( t=1 GDPt GDP ) 2 To asses the probability limit of the bias term we need the reduced form: GDP t = a + µ I 1 b + ɛ Ct 1 b + ɛ It 1 b, for 0 < b < 1 C t = (a + bµ I ) 1 b + ɛ Ct 1 b + b 1 b ɛ It which together with the assumptions give the properties of the two random variables GDP t and C t 17 / 20

18 Simultaneous equations bias I plim(ˆb b) = plim T t=1 GDP t (ɛ Ct ɛ C ) T ( t=1 GDPt GDP ) 2 plim(ˆb b) = plim T t=1 From the assumptions of the model: ( ) a+µi 1 b + ɛ Ct 1 b + ɛ It 1 b T t=1 ( ɛct ɛ C 1 b + ɛ It ɛ I 1 b (ɛ Ct ɛ C ) ) 2 plim(ˆb b) = σ2 C σ 2 C + σ2 I > 0 (20) 18 / 20

19 Simultaneous equations bias II OLS is overestimating the structural parameter b in the model given by (14)-(19) Trygve Haavelmo: The statistical implications of system of simultaneous equations, Econometrica (1943) Applies to parameter estimation of all simultaneous equations models: Market supply and demand equations for example, See HGL Ch / 20

20 Summary The relationship C t = a + b(gdp t ) + ɛ Ct looks like an ordinary regression. But OLS estimators are not consistent. What is going on? The point is that, in the model (14)-(19) the consumption function (14) is not a conditional expectation, and therefore the assertion E (ɛ Ct GDP t ) = 0 cannot be made. (14) is a linear in parameters econometric relationship but it is is not a conditional expectation. We say that the parameter of interest (b) is in this case, not a parameter in the conditional expectation function for C t given GDP t. Lecture 17: Another example of an econometric relationship which is not a conditional expectation. Then: estimation methods that give consistent estimators of b in (14)-(19). 20 / 20