Principles of forecasting

Transcription

1 2.5 Forecasting

2 Principles of forecasting Forecast based on conditional expectations Suppose we are interested in forecasting the value of y t+1 based on a set of variables X t (m 1 vector). Let y t+1 t denote such a forecast. To evaluate the usefulness of the forecast we need to specify a loss function. Here we specify a quadratic loss function. A quadratic loss function means that y t+1 t is chose to minimize E(y t+1 y t+1 t ) 2. E(y t+1 y t+1 t ) 2 denoted is known as the mean squared error associated with the forecast y t+1 t MSE(y t+1 t )=E(y t+1 y t+1 t ) 2

3 Fundamental result: the forecast with the smallest MSE is the expectation of y t+1 t conditional on X t that is y t+1 t = E(y t+1 X t ) We now verify the claim. Let g(x t ) be any other function and let y t+1 t = g(x t ). The associated MSE is Define E [y t+1 g(x t )] 2 = E [y t+1 E(y t+1 X t )+E(y t+1 X t ) g(x t )] 2 = E [y t+1 E(y t+1 X t )] E {[y t+1 E(y t+1 X t )][E(y t+1 X t ) g(x t )]} + + [E(y t+1 X t ) g(x t )] 2 η t+1 [y t+1 E(y t+1 X t )][E(y t+1 X t ) g(x t )] (33) The conditional expectation is E(η t+1 X t ) = E {[y t+1 E(y t+1 X t )][E(y t+1 X t ) g(x t )] X t } = [E(y t+1 X t ) g(x t )] E {[y t+1 E(y t+1 X t )] X t } = [E(y t+1 X t ) g(x t )] [E(y t+1 X t ) E(y t+1 X t )] = 0

4 Therefore by law of iterated expectations This means that E(η t+1 )=E(E(η t+1 X t )) = 0 E [y t+1 g(x t )] 2 = E [y t+1 E(y t+1 X t )] 2 + E([E(y t+1 X t ) g(x t )]) 2 Therefore the function that minimizes the MSE is g(x t )=E(y t+1 X t ) E(y t+1 X t ) is the optimal forecast of Y t+1 conditional of X t under a quadratic loss function. The MSE of this optimal forecast is E[y t+1 g(x t )] 2 = E[y t+1 E(y t+1 X t )] 2

5 Forecast based on linear projections We now restrict the class of forecasts we consider to be linear function of X t : y t+1 t = α X t Suppose α is such that the resulting forecast error is uncorrelated with X t E[(y t+1 α X t )X t] = 0 (34) If (34) holds the we call α X t the linear projection of y t+1 on X t. The linear projection produces the smallest mean square forecast error among the class of linear forecasting rules. To verify this let g X t be any arbitrary forecasting rule. the middle term E ( y t+1 g X t ) 2 = E ( y t+1 α X t + α X t g X t ) 2 = E ( y t+1 α X t ) 2 + [( )( )] +2E y t+1 α X t α X t g X t + ( ) +E α X t g 2 X t E[ ( y t+1 α X t )( α X t g X t ) ] = E[ ( yt+1 α X t ) X t ] [α g] by definition of linear projection. Thus = 0 E ( y t+1 g X t ) 2 = E ( y t+1 α X t ) 2 + E ( α X t g X t ) 2 The optimal linear forecast is the value g X t = α X t.

6 We use the notation P (y t+1 t )=α X t to indicate the linear projection of y t+1 on X t. Notice that MSE[P (y t+1 X t )] MSE [E(y t+1 X t )] The projection coefficient α can be calculated in terms of moments of y t+1 and X t. E(y t+1 X t )=α E(X t X t ) α = E(y t+1 X t )[E(X tx t )] 1 It is easy to generalize to the multivariate case. Let Y t be a n 1 vector α an m n matrix and X t an m 1 vector. The linear pojection is defined to be the vactor α X t satisfying E(Y t+1 α X t )X t =0

7 Linear projections and OLS regression There is a close relationship between OLS estimator and the linear projection coefficient. If Y t+1 and X t are stationary processes and also ergodic for the second moments then implying (1/T ) (1/T ) T t=1 T t=1 X t X t p E(X t X t) X t y t+1 p E(Xt y t+1 ) ˆβ p α The OLS regression yields a consistent estimate of the linear projection coefficient.

8 Forecasting with VAR models Iterated forecast Let us consider the VAR(p) in companion form Y t = AY t 1 + e t (35) where e t is white noise. The h-step ahead linear predictor of Y t+h conditional on the information available at time t, Y t+h t, is given by Y t+h t = A h Y t = AY t+h 1 t (36) where the first n rows of Y t+h t represent the optimal forecast of Y t+h. From (36) it is easy to compute recursively the forecast for Y t+h at any horizon. The predictor in the previous slide is optimal in the sense that, as we know, it delivers the minimum MSE among those that are linear functions of Y.

9 Using we get the forecast error Y t+h = A h Y t + Y t+h Y t+h t = h 1 i=0 h 1 i=0 A i e t+h i (37) A i e t+h i (38) From the forecast error it is easy to obtain the Mean Square Error, the covariance of the forecast error, MSE[Y t+h t ]=E ( Y t+k Y t+h t ) (Yt+k Y t+h t ) =Σ(h) = h 1 i=0 A i ΩA i = Σ(h 1) + A h 1 ΩA h 1 the MSE for will be the first upper left n n matrix. Notice that the MSE is non decreasing and that as h will approach the variance of Y t.

10 Direct forecast An alternative is to compute the direct forecast by computing the projection of Y t+h on Y t. To see this consider a bivariate VAR(p) with two variables, x t and y t. We want to forecast x t+h given the incormation available at time t. The direct forecast works as follows: 1. Estimate the projection equation x t = a + p 1 φ i x t h i + p 1 i=0 i=0 ψ i y t h i + ε t 2. Using the estimated coefficients, the predictor x t+h t is obtain as ˆx t+h t = a + p 1 φ i x t i + p 1 i=0 i=0 ψ i y t i

11 Forecast evaluation: pseudo out-of-sample exercises A key issue in forecasting is to evaluate the forecasting accuracy of a model of interest. In particular several times we will be interested in comparing the performance of competing forecasting models. How can we perform such a forecast evaluation? Answer: we can compare the mean squared errors using pseudo out-of-sample forecast exercises.

12 Suppose we have a sample of T observations. Let τ = T 0 <T. A pseudo out-of-sample exercise works as follows: 1. We use τ observations to estimate the parameters of the model. 2. We forecast Y τ+j with Ŷ τ+j τ j =1, 2,..., s. 3. We compute the forecast error w τ+j τ = Y τ+j Ŷ τ+j τ with j =1, 2,..., s. 4. We update τ = τ + 1 and repeat steps 1-3. We repeat steps 1-4 up to the end of the sample and we compute the mean squared error for each variable i =1,...n. MSE(Ŷ i,t +j T )= 1 T T 0 j T T 0 j τ=t 0 w 2 i,τ+j τ or the root mean squared error RMSE(Ŷ i,t +j T )= 1 T T 0 j T T 0 j τ=t 0 w 2 i,τ+j τ Repeating the same exercise for various models we can choose the one which has the smallest MSE or RMSE.

13 Application: Forecasting inflation and unemployment D Agostino Gambetti and Giannone (JAE 2012) consider a trivariate VAR(2) model including inflation unemployment and the short term interest rate. The sample spans from 1948:I-2007:IV. Forecast up to 12 quarters ahead. Estimation is in real time and is made using both VAR and AR for each of the series. Estimation is done both recursively snd with rolling window.

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