Econ 4120 Applied Forecasting Methods L10: Forecasting with Regression Models. Sung Y. Park CUHK

Transcription

1 Econ 4120 Applied Forecasting Methods L10: Forecasting with Regression Models Sung Y. Park CUHK

2 Conditional forecasting model Forecast a variable conditional on assumptions about other variables. (scenario analysis) y T +h,t x T +h = β 0 + β 1 x T +h, where x T +h is the h-step ahead value of x. Assuming normality, the conditional density forecast N(y T +h,t x T +h, σ2 ), and conditional interval forecast follow immediately.

3 Parameter uncertainty Forecasts are subject to errors: three major sources specification uncertainty innovation uncertainty parameter uncertainty Specification and innovation uncertainties are more important than parameter uncertainty. In most case, it is a little hard to quantify the effect of parameter uncertainty.

4 Parameter uncertainty Consider a simple case: y t = βx t + ε t If x T +h = x T +h, y T +h = βx T +h + ε T +h ŷ T +h x T +h = ˆβx T +h. ê T +h,t = y T +h ŷ T +h x T +h = (β ˆβ)x T +h + ε T +h Var(ê T +h,t ) = x T +h 2 Var( ˆβ) + σ 2. We can easily show that Var( ˆβ) = σ 2 / T t=1 x t 2. Thus T Var(ê T +h,t ) = (σ 2 / xt 2 )xt +h 2 + σ 2. t=1

5 Parameter uncertainty T Var(ê T +h,t ) = (σ 2 / xt 2 )xt +h 2 + σ 2. t=1 (σ 2 / T t=1 x 2 t )x T +h 2 : parameter uncertainty σ 2 : innovation uncertainty Density forecast: N ˆβx T +h 2, ) ˆσ 2 T t=1 x x t 2 T +h 2 + ˆσ 2

6 Unconditional forecasting model We do not forecast of y conditional on assumptions about x: find the best possible forecast of y. We may need to forecast the right-hand-side variables. y T +h,t = β 0 + β 1 x T +h,t, Without a forecast of x the model at hand does not help us. We can use ˆx T +h,t to forecast y. Or y t = β 0 + δx t 1 + ε t for one-step-ahead forecast. (What if we forecast 2-step-ahead...)

7 Distributed Lags y t = β 0 + δx t 1 + ε t N x y t = β 0 + δ i x t i + ε t i=1 y depends on a distributed lag of past x s. δ s are lag weights and their pattern is called the lag distribution. Some case N x is too large : loose many degree of freedom. We can consider the following model: subject to min β 0,δ i T t=n x +1 δ i = a + bi + ci 2, ) N 2 x y t β 0 δ i x t i. i=1 i = 1, 2,, N x

8 Rational distributed lags Why should the lag weights follow a low order polynomial? Rational distributed lags y t = A(L) B(L) x t + ε t, where A(L) and B(L) are low-order polynomials. B(L)y t = A(L)x t + B(L)ε t

9 ADL Model Consider N x y t = β 0 + δ i x t i + ε t Something is missing here! The dependent variable is highly likely correlated with to its own past... N y i=1 N x y t = β 0 + α i y t i + δ i x t i + ε t i=1 a distributed lag regression model with lagged dependent variables (or autoregressive distributed lag model). (This is not the same as rational distributed lag model...) i=1

10 ADL Model One can also consider a distributed lag regression model with ARMA disturbance N x y t = β 0 + δ i x t i + ε t i=1 ε t = Θ(L) Φ(L) v t v t WN(0, σ 2 ) Regressions with ARMA disturbances are very traditional ways in statistical and econometric tools.

11 ADL Model distributed lag regression model with autoregressive disturbances: y t = β 0 + β 1 x t 1 + ε t ε t = φε t 1 + v t v t WN(0, σ 2 ) The above model is the same as (why?) y t = φy t 1 + (1 φ)β 0 + β 1 x t 1 φβ 1 x t 2 + v t subject to the restriction on φ and β 1. Notes: Distributed lag regressions with lagged dependent variables are more general than distributed lag regressions with dynamic disturbances.

12 Transfer function models Consider y t = A(L) B(L) x t + C(L) D(L) ε t This model can capture cross-variable dynamics. Many models are special cases of the above model.

13 Forecasting with Regression Models 227 Now multiply both sides by (1 cpl) to get or (1 - vl)y, = (1 - cp)po + Pi (1 - q> )*i-i + v, y, = cpj,-i + (1 - cp)(3 0 + Pi»,-i - <p0i**-2 + "< Thus, a model with one lag of x on the right and AR(1) disturbances is equivalent to a model with y t -i, «*-i, and x t -2 on the right-hand side and white noise errors, subject to the restriction that the coefficient on the second lag of x,_ 2 is the negative of the product of the coefficients on y t _\ and x,-\. Distributed lag regressions with lagged dependent variables are more general than distributed lag regressions with dynamic disturbances. Transfer function models are more general still and include both as special cases. 4 The basic idea is to exploit the power and parsimony of rational distributed lags in modeling both own-variable and cross-variable dynamics. Imagine beginning with a univariate ARMA model, yi C{L) D(L) e, Name Model Restrictions Transfer function Standard distributed lag Rational distributed lag A(L) C(L) 3y, = x, H / B(L) D(L) y, = A(L)x, +e ; _ ML) yi ~~B(L) X ' +Z ' None B(L) = C(L) = D(L) = 1 C(L) = D(L) = 1 TABLE 11.1 The Transfer Function Model and Various Special Cases Univariate AR Univariate MA Univariate ARMA Distributed lag with lagged dependent variables Distributed lag with ARMA disturbances Distributed lag with AR disturbances 1 ft = E( A(L) = 0, C(L) = 1 3 D(L) A(L) =0,D(L) = 1 y, = C(L)e, A{L) = 0 C(L) p "it = / 3 D{L) B(L)y, = A(L)x, +,,or e t C(L) = 1,D(L) = B(L) _ A(L) B(L). B{L) 1 Jt TT7TT x i C{L) yt = A{L) Xt+w - ) Z i B(L) y, = A(L)x, + D(L) B(L) = C(L) = 1 4 Table 11.1 displays a variety of important forecasting models, all of which are special cases of the transfer function model.