Model Averaging in Predictive Regressions
|
|
- Jeremy Jones
- 5 years ago
- Views:
Transcription
1 Model Averaging in Predictive Regressions Chu-An Liu and Biing-Shen Kuo Academia Sinica and National Chengchi University Mar 7, 206 Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, 206 / 48
2 Introduction Model uncertainty: the challenge of empirical studies is that one does not know exactly what predictors should be included in the model. Two methods to deal with model uncertainty: model selection and model averaging. Model averaging: a weighted average of estimates from candidate models. Two model averaging approaches: Bayesian model averaging and Frequentist model averaging. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
3 Introduction Since the seminal work of Bates and Granger (969), forecast combination has been widely used in economics and statistics. How to form the forecast weights is still an open question. Many methods have been proposed for forecast combination, including Granger and Ramanathan (984), Min and Zellner (993), Raftery, Madigan, and Hoeting (997), Buckland, Burnham, and Augustin (997), Yang (2004), Hansen (2008), Elliott, Gargano, and Timmermann (203), and Cheng and Hansen (205), among others. We propose a new method for weight selection for linear models. We do not impose the i.i.d. normal assumption. The set of candidate models could be nested or non-nested. The proposed averaging criterion is an an asymptotically unbiased estimator of the mean squared forecast error. We balance the trade-off between model biases and estimation variances. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
4 Model and Estimation We deal with the one-step-ahead forecasting model: h t = (x t,z t ). y t+ = x t β +z t γ +e t+, E(h t e t+ ) = 0. x t is a set of must-have predictors z t is a set of potentially relevant predictors. It could be lags of y t, deterministic terms, or the interaction terms between the predictors. The error term is allowed to be heteroskedastic. The goal is to construct a point forecast of y T+ given (x T,z T ). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
5 Approximating Models Submodel: The mth model includes all must-have predictors x t and a subset of potentially relevant predictors z t. The mth model has p +q m predictors for m =,...,M. The set of models could be nested or non-nested. The least-squares estimator of β in the mth model is θ m = (H m H m) H m y, where H = (X,Z), H m = (X,Z m ) = HS m, and S m is a (p +q) (p +q m ) selection matrix. The predicted value is ŷ(m) = H m θm = HS m θm. The one-step-ahead forecast is ŷ T+ T (m) = h T S m θ m. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
6 One-Step-Ahead Combination Forecast Let w = (w,...,w M ) be a weight vector with w m 0 and M { m= w m =. That is, w H M where H M = w [0,] M : } M m= w m =. The one-step-ahead combination forecast is ŷ T+ T (w) = = M w m ŷ T+ T (m) m= M w m h TS m θm m= = h T θ(w) where θ(w) = M m= w ms m θm is an averaging estimator of θ. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
7 Question and Contributions Question: How to assign the model weights? Contributions: Optimal Weights: Show that the optimal model weights that minimize the mean squared forecast error (MSFE) depend on the local parameters and the covariance matrix of the predictive regression. Data-Driven Weights: Propose a plug-in estimator of the infeasible optimal weights and use these estimated weights to construct the forecast combination. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
8 Outline Outline Asymptotic Risk, MSE and MSFE 2 Weight Selection 3 Finite Sample Investigation 4 Empirical Application 5 Multi-Step Forecast Combination Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
9 Asymptotic Risk, MSE and MSFE Outline Asymptotic Risk, MSE and MSFE 2 Weight Selection 3 Finite Sample Investigation 4 Empirical Application 5 Multi-Step Forecast Combination Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
10 Asymptotic Risk, MSE and MSFE MSE and MSFE Goal: Select weights to minimize the one-step-ahead MSFE. Let σ 2 = E(et 2) and µ t = x t β +z tγ be the conditional mean. The in-sample mean squared error (MSE): ( ) T MSE(w) = E (µ t µ t(w)) 2. T t= The one-step-ahead mean squared forecast error: MSFE(w) = E ( y T+ ŷ T+ T (w) ) 2 = E (e T+ 2 +(µ T µ T (w)) 2) E (e T+ 2 +(µ t µ t(w)) 2) = σ 2 +MSE(w). Therefore the optimal weight vector that minimizes the MSE(w) is expected to minimizes the MSFE(w). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
11 Asymptotic Risk, MSE and MSFE Asymptotic Risk How to approximate the MSE? Use the asymptotic risk to approximate the MSE. Let Q = E(h th t). Define the asymptotic trimmed risk or weighted MSE of an estimator θ for θ as R( θ,θ) = lim ζ liminf n Emin{T( θ θ) Q( θ θ),ζ}. The weighted MSE function plus σ 2 corresponds to one-step-ahead MSFE. Use the information from the sum of squared errors. Define P(w) = M m= wmhm(h mh m) H m and ê(w) = y Hˆθ(w). Then, we have E(ê(w) ê(w)) = MSE(w)+Tσ 2 2E(e P(w)e). Mallows criterion (Hansen, 2007): C T (w) = ê(w) ê(w)+2σ 2 k w where k = (k,...,k M ). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, 206 / 48
12 Weight Selection Outline Asymptotic Risk, MSE and MSFE 2 Weight Selection 3 Finite Sample Investigation 4 Empirical Application 5 Multi-Step Forecast Combination Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
13 Weight Selection Local Asymptotic Framework We follow Hjort and Claeskens (2003, JASA) and use a local-to-zero asymptotic framework to approximate the MSE. Assumption. γ = γ T = δ/ T, where δ is an unknown constant vector. The local-to-zero framework is canonical in the sense that both squared model biases and estimator variances have the same order O(T ). We can decompose ˆθ m as ˆθ m = θ m +(H m H m) H m Z(I q Π m Π m)γ T +(H m H m) H m e where θ m = (β,γ m ) and Π m is a q m q selection matrix. (I q Π mπ m) is the selection matrix that chooses the omitted auxiliary regressors. If γ T converges to 0 slower than T /2, the asymptotic bias goes to infinity. If γ T converges to 0 faster than T /2, the asymptotic bias goes to zero. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
14 Weight Selection Local Asymptotic Framework Assumption 2. {y t+,h t } is a strictly stationary and ergodic time series with finite r > 4 moments and E(e t+ F t ) = 0, where F t = σ(h t,h t,...;e t,e t,...). Assumption 2 states that data is strictly stationary. It implies that e t+ is conditionally unpredictable at time t. It is sufficient to imply that where Ω = E ( h t h te 2 t+). T H H p Q T /2 H e d R N(0,Ω) Also, we have T H mh m p Q m where Q m = S mqs m is nonsingular. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
15 Weight Selection Asymptotic Normality of the Averaging Estimator Theorem. Suppose that Assumptions 2 hold. As T, we have T ( θ(w) θ ) d N ( A(w)δ,V(w) ) A(w) = V(w) = M w m (P m Q I p+q )S 0 m= M wm 2 P mωp m +2 w m w l P m ΩP l m= where P m = S m Q m S m and S 0 = (0 q p,i q ). m l Remark: A(w)δ represents the bias term. The magnitude of the bias is determined by the covariance matrix Q and the local parameter δ. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
16 Weight Selection Asymptotic Trimmed Risk of the Averaging Estimator We derive the asymptotic trimmed risk of the model averaging estimator and characterize the optimal weights in a local asymptotic framework. Theorem 2. Suppose Assumptions -2 hold. We have R( θ(w),θ) = w ψw where ψ is an M M matrix with the (m,l)th element ψ m,l = tr(qc m δδ C l )+tr(qp mωp l ). Note that C m and P m are functions of Q and the selection matrix. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
17 Weight Selection Optimal Weights and Plug-In Weights The optimal weight vector is the value that minimizes the asymptotic risk over w H M : w o = argmin w H M w ψw. The weight vector of the plug-in estimator is defined as ŵ = argmin w H M w ψw, where w ψw is the sample analog of w ψw. The objection function is linear-quadratic in w, which can be solved numerically via quadratic programming. The plug-in one-step-ahead combination forecast is ŷ T+ T (ŵ) = h T θ(ŵ). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
18 Weight Selection Construct the Data-Driven Weights We now discuss the plug-in estimator ψ m,l. Recall that ψ m,l = tr(qc m δδ C l )+tr(qp mωp l ). We use the method of moments estimator for covariance matrices Q and Ω. Thus, it is quite easy to model the heteroskedasticity and serial correlation. We use the asymptotically unbiased estimator for the local parameter δ. δ = T γ d R δ N(δ,S 0Q ΩQ S 0). An alternative estimator is δδ = δ δ S 0 Q Ω Q S 0. The asymptotic trimmed risk of the plug-in averaging estimator: ŵ d w = argmin w H M w ψ w where ψ m,l = tr(qc mr δ R δc l)+tr(qp mωp l ). R( θ(ŵ),θ) = E((A(w )δ +P(w )R) Q(A(w )δ +P(w )R)). A(w ) = M m= w m (P mq I p+q)s 0 and P(w ) = M m= w mp m. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
19 Finite Sample Investigation Outline Asymptotic Risk, MSE and MSFE 2 Weight Selection 3 Finite Sample Investigation 4 Empirical Application 5 Multi-Step Forecast Combination Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
20 Finite Sample Investigation Finite Sample Investigation We consider two simulation setups. The first design is the regression model and we consider all possible models. The second design is a moving average model with exogenous inputs and we consider a sequence of nested candidate models. We consider the following estimators: () Smoothed AIC (S-AIC; Buckland, Burnham, and Augustin (997)) (2) Smoothed BIC (S-BIC) (3) Mallows model averaging (MMA; Hansen (2007)) (4) Jackknife model averaging (JMA; Hansen and Racine (202)) (5) Complete subset regression (Elliott, Gargano, and Timmermann (203)) (6) Plug-In averaging estimators Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
21 Finite Sample Investigation Six Forecast Combination Methods Weight choice: Remark: S-AIC: ŵ m = exp( 2 AIC m)/ M j= exp( 2 AIC j). S-BIC: ŵ m = exp( 2 BIC m)/ M j= exp( 2 BIC j). MMA: C T (w) = ê(w) ê(w)+2σ 2 k w JMA: CV n (w) = w ẽ ẽw where ẽ = (ẽ,...,ẽ M ) is the T M matrix of leave-one-out least squares residuals. MMA is limited to the homoskedastic model. Both MMA and JMA are limited to the random sample. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
22 Finite Sample Investigation Complete Subset Regressions Elliott, Gargano, and Timmermann (203, JoE) propose a new forecast combination method based on complete subset regressions. For a given set of potential predictors, they construct the forecast combination by using equal-weighted combination based on all possible models that include κ predictors. The one-step-ahead combination forecast is ŷ T+ T (κ) = n κ,k n κ,k h T S m θ m s.t. tr(s m S m ) = κ, m= where n κ,k = k!/(κ!(k κ)!) is the number of models considered based on κ subset regressions. Remark: Complete subset regressions is not suitable for the nested models. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
23 Finite Sample Investigation DGP : Regression Model The data generation process for the first design is k y t+ = β j x jt +e t+, j= x jt = ρ x x jt +u jt, for j 2. x t is the intercept. x jt for j 2 are AR() processes with ρ x = 0.5, and 0.9. The predictors x jt are correlated. (u 2t,...,u kt ) N(0,Q u ) where the diagonal elements of Q u are, and off-diagonal elements are ρ u. We set ρ u = 0.25,0.5,0.75, and 0.9. β = (, k k,..., k) c/ T and δj = Tβ j = c(k j +)/k. Homoskedastic simulation: e t N(0,). Heteroskedastic simulation: e t = 3 /2 ( ρ 2 x )x2 kt ǫ t and ǫ t follows AR(). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
24 Finite Sample Investigation DGP : Regression Model The sample size is T = 200. The number of predictors is k = 5. The number of models is M = 32. We report the relative risk: S S s= S min m {,...,M} S s= ( ys,t+ T ŷ s,t+ T (ŵ) ) 2 ( ys,t+ T ŷ s,t+ T (m) ) 2, where ŷ T+ T (m) is the prediction based on the model m and ŷ T+ T (ŵ) is the prediction based on the averaging estimator. (S = 5000) Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
25 Finite Sample Investigation Heteroskedastic simulation, ρ x = ρ u = 0.25 S BIC S AIC MMA JMA Plug In R 2 Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
26 Finite Sample Investigation Heteroskedastic simulation, ρ x = 0.9. ρ u = 0.25 ρ u = S BIC S AIC MMA JMA Plug In R R 2 ρ u = 0.75 ρ u = R R 2 Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
27 Finite Sample Investigation Homoskedastic simulation, ρ x = 0.9. ρ u = 0.25 ρ u = S BIC S AIC MMA JMA Plug In R R 2 ρ u = 0.75 ρ u = R R 2 Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
28 Finite Sample Investigation Simulation Results Monte Carlo simulations show that the plug-in averaging estimtor has much lower MSFE than other model averaging estimators in both homoskedastic and heteroskedastic settings. JMA has lower reative risk than MMA and S-AIC in the heteroskedastic simulation. S-BIC has poor performance in both homoskedastic and heteroskedastic settings. We now compare the plug-in averaging estimator with complete subset regressions. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
29 Finite Sample Investigation Homoskedastic simulation, ρ x = 0.5. ρ u = 0.25 ρ u = κ= κ=2 κ=3 κ=4 κ=5 Plug In R R 2 ρ u = 0.75 ρ u = R R 2 Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
30 Finite Sample Investigation Heteroskedastic simulation, ρ x = 0.5. ρ u = 0.25 ρ u = κ= κ=2 κ=3 κ=4 κ=5 Plug In R R 2 ρ u = 0.75 ρ u = R R 2 Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
31 Finite Sample Investigation DGP 2: MAX(, ) The data generation process for the second design is y t = x t +0.5x t +e t +βe t, x t = 0.5x t +u t. x t is an AR() process and u t N(0,). e t N(0,σ 2 t), where σ 2 t = 0.5 for the homoskedastic simulation and σ 2 t = +x 2 t for the heteroskedastic simulation. The parameter β is varied on a grid from 0.5 to 0.5. We consider a sequence of nested models based on regressors {,y t,x t,y t 2,x t,y t 3,x t 2 }. For β 0, the true model is infinite dimensional. For β = 0, all seven models are wrong. Sample size: T = 00, 200, 500, and 000. We report the relative risk and the average model size. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
32 Finite Sample Investigation Relative risk, homoskedastic errors T = T = 200 S BIC S AIC MMA JMA Plug In β β.6 T = T = β β Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
33 Finite Sample Investigation Relative risk, heteroskedastic errors T = T = 200 S BIC S AIC MMA JMA Plug In β β.6 T = T = β β Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
34 Finite Sample Investigation Model size, homoskedastic errors. 7 T = 00 7 T = 200 S-BIC S-AIC MMA Model Size 6 5 Model Size 6 5 JMA Plug-In β β 7 T = T = 000 Model Size 6 5 Model Size β β Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
35 Finite Sample Investigation Model size, heteroskedastic errors. 7 T = 00 7 T = 200 S-BIC S-AIC Model Size 6 5 Model Size 6 5 MMA JMA Plug-In β β 7 T = T = Model Size 5 Model Size β β Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
36 Empirical Application Outline Asymptotic Risk, MSE and MSFE 2 Weight Selection 3 Finite Sample Investigation 4 Empirical Application 5 Multi-Step Forecast Combination Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
37 Empirical Application Empirical Application: Stock Returns Prediction We apply the plug-in forecast combination method to stock returns prediction. Different studies suggest different economic variables and models. Welch and Goyal (2008) argue that numerous economic variables have poor out-of-sample predictions. Rapach, Strauss, and Zhou (200) propose a equal-weighted forecast combination approach to the subset predictive regression. We apply the forecast combination with data-driven weights instead of equal weights to U.S. stock market. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
38 Empirical Application Model and Data The model: r t+ = β +z tγ +e t+ r t+ is the equity premium and z t are the economic variables. We consider 0 economic variables and all possible models. The 0 economic variables are Dividend Price Ratio, Dividend Yield, Earnings Price Ratio, Book-to-Market Ratio, Net Equity Expansion, Treasure Bill, Long Term Return, Default Yield Spread, Default Return Spread, and Inflation. The quarterly data are from Welch and Goyal (2008) for (T=260). We follow Welch and Goyal (2008) and calculate the out-of-sample forecast of the equity premium using a recursively expanding estimation window. In-sample period: 947:-964:4 Out-of-sample evaluation period: 965:-20:4 We use the historical average of the equity premium as a benchmark. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
39 Empirical Application The differences between the cumulative square prediction errors of the historical average forecasting model and the cumulative square prediction errors of the forecast combination model for 965:-20: S AIC S BIC MMA JMA RSZ (200) Plug In Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
40 Empirical Application The differences between the cumulative square prediction errors of the historical average forecasting model and the cumulative square prediction errors of the forecast combination model for 965:-20: κ= 0.06 κ=6 κ=2 κ= κ=3 κ= κ=8 κ=9 0. κ=5 Plug In κ=0 Plug In Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
41 Empirical Application Out-Of-Sample Forecasting Results The out-of-sample R 2 value of the plug-in averaging estimator is with the p-value The out-of-sample R 2 value is computed as R 2 OOS = T τ=τ 0 ( rτ+ r τ+ τ (ŵ) ) 2 T τ=τ 0 ( rτ+ r τ+ τ ) 2 where r τ+ τ = τ t= rt is the historical average and r T+ T(ŵ) is the equity premium forecast based on forecast combination. The associated p-value is based on Clark and West (2007) to test the null hypothesis that R 2 OOS 0. Our results support that forecast combinations provide significant gains on equity premium predictions relative to the historical average. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
42 Multi-Step Forecast Combination Outline Asymptotic Risk, MSE and MSFE 2 Weight Selection 3 Finite Sample Investigation 4 Empirical Application 5 Multi-Step Forecast Combination Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
43 Multi-Step Forecast Combination Multi-Step Forecast: Model and Estimation We now consider the h-step-ahead forecasting model: y t+h = x t β +z t γ +e t+h E(h t e t+h ) = 0. The goal is to construct a point forecast of y T+h given (x t,z t ). The h-step-ahead forecast from the mth model is ŷ T+h T (m) = h TS m θm, where θ = (H H) H y. The h-step-ahead combination forecast is where θ(w) = M m= w ms m θm. ŷ T+h T (w) = h T θ(w), Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
44 Multi-Step Forecast Combination Optimal Weights and Plug-In Weights We now modify Assumption 2 as follows: Assumption 2. {y t+h,h t } is a strictly stationary and ergodic time series with finite r > 4 moments and E(e t+h F t ) = 0, where F t = σ(h t,h t,...;e t,e t,...). Suppose that Assumptions and 2 hold. Then the results in Theorems 3 still hold and the optimal weight vector has the same form w o = argmin w H M w ψw where the (m,l)th element of ψ is ψ m,l = tr(qc m δδ C l )+tr(qp mωp l ) and Ω = lim T T T s= T t= E(h sh t e s+he t+h ). We also can construct the plug-in estimator by replacing the unknown paramters Q, Ω, and δ by the sample analogue. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
45 Multi-Step Forecast Combination Relationship between the Plug-In Averaging Estimator and the Mallows C p -type Averaging Estimator Suppose that there is no must-have predictor, i.e., x t is an empty matrix. Then we have S m = Π m, S 0 = I q, and Ĉm = P m Q Iq. The plug-in estimator can be rewritten as ψ m,l = tr( QĈm( δ δ Q Ω Q )Ĉ l)+tr( Q P m Ω Pl ) ) = tr( Q( Pm Q Iq ) δ δ ( Q P l I q ) tr ( Q( Pm Q Iq ) Q Ω Q ( Q P l I q ) Q P ) m Ω Pl = (ê mêl ê ê)+tr( Q m Ω m )+tr( Q Ω l ) tr( Q Ω), where ê = y H θ, ê m = y H m θm, Q m = S m QS m, and Ω m = S m ΩS m. l Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
46 Multi-Step Forecast Combination The Equivalent Result The criterion function for the plug-in averaging estimator is The (m,l)th element of ψ is w ψw = w ψw ê ê tr( Q Ω). ψ m,l = ê mêl +tr( Q Ω m m )+tr( Q Ω l ). Minimizing w ψw over w = (w,...,w M ) is equivalent to minimizing w ψw. If the error term is i.i.d. and homoskedastic, then the covariance matrix Ω can be consistently estimated by Ω = σ 2 Q. Thus, tr( Q m Ω m ) = σ 2 k m. Define Σ as an M M matrix whose (m,l)th element is k m +k l. The criterion function for the plug-in averaging estimator is w ψw = ê(w) ê(w)+ σ 2 w Σw = ê(w) ê(w)+2 σ 2 k w which is the Mallows criterion proposed by Hansen (2007). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48 l
47 Multi-Step Forecast Combination The Equivalent Result 2 If the error term is serially uncorrelated and identically distributed, then Ω can be consistently estimated by Ω = T T t= h th tê2 t+. ( ( T Thus, tr( Q Ω ) ( T m m ) = tr t= h m,t t+) ) h m,tê2 k m. t= h m,t h m,t Define Σ as an M M matrix whose (m,l)th element is k m + k l. The criterion function for the plug-in averaging estimator is where k = ( k,..., k M ). w ψw = ê(w) ê(w)+w Σw = ê(w) ê(w)+2 k w, This is equivalent to the heteroskedasticity-robust C p criterion proposed by Liu and Okui (203). Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
48 Multi-Step Forecast Combination Conclusion We study the weight selection for forecast combination in a predictive regression when the goal is minimizing the MSFE. We derive the asymptotic distribution and asymptotic risk of the averaging estimator in a local asymptotic framework without the i.i.d. normal assumption. We propose a frequentist model averaging criterion, an asymptotically unbiased estimator of the asymptotic risk, to select forecast weights. Simulations show that the proposed estimator achieves lower MSFE relative risk than other existing model averaging methods in most cases. The proposed method can be easily extended to the multi-step forecast combination. Liu & Kuo (IEAS & NCCU) Model Averaging in Predictive Regressions Mar 7, / 48
Model Averaging in Predictive Regressions
MPRA Munich Personal RePEc Archive Model Averaging in Predictive Regressions Chu-An Liu and Biing-Shen Kuo Academia Sinica, National Chengchi University 8 March 206 Online at https://mpra.ub.uni-muenchen.de/706/
More informationLeast Squares Model Averaging. Bruce E. Hansen University of Wisconsin. January 2006 Revised: August 2006
Least Squares Model Averaging Bruce E. Hansen University of Wisconsin January 2006 Revised: August 2006 Introduction This paper developes a model averaging estimator for linear regression. Model averaging
More informationOn the equivalence of confidence interval estimation based on frequentist model averaging and least-squares of the full model in linear regression
Working Paper 2016:1 Department of Statistics On the equivalence of confidence interval estimation based on frequentist model averaging and least-squares of the full model in linear regression Sebastian
More informationUsing all observations when forecasting under structural breaks
Using all observations when forecasting under structural breaks Stanislav Anatolyev New Economic School Victor Kitov Moscow State University December 2007 Abstract We extend the idea of the trade-off window
More informationESSAYS ON MODEL AVERAGING. Chu-An Liu. A dissertation submitted in partial fulfillment of the requirements for the degree of. Doctor of Philosophy
ESSAYS ON MODEL AVERAGING by Chu-An Liu A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Economics) at the UNIVERSITY OF WISCONSIN MADISON 0 Date
More informationComparing Forecast Accuracy of Different Models for Prices of Metal Commodities
Comparing Forecast Accuracy of Different Models for Prices of Metal Commodities João Victor Issler (FGV) and Claudia F. Rodrigues (VALE) August, 2012 J.V. Issler and C.F. Rodrigues () Forecast Models for
More informationJackknife Model Averaging for Quantile Regressions
Singapore Management University Institutional Knowledge at Singapore Management University Research Collection School Of Economics School of Economics -3 Jackknife Model Averaging for Quantile Regressions
More informationForecasting Lecture 2: Forecast Combination, Multi-Step Forecasts
Forecasting Lecture 2: Forecast Combination, Multi-Step Forecasts Bruce E. Hansen Central Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts
More informationComplete Subset Regressions
Complete Subset Regressions Graham Elliott UC San Diego Antonio Gargano Bocconi University, visiting UCSD November 7, 22 Allan Timmermann UC San Diego Abstract This paper proposes a new method for combining
More informationJournal of Econometrics
Journal of Econometrics 46 008 34 350 Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom Least-squares forecast averaging Bruce E. Hansen
More informationJackknife Model Averaging for Quantile Regressions
Jackknife Model Averaging for Quantile Regressions Xun Lu and Liangjun Su Department of Economics, Hong Kong University of Science & Technology School of Economics, Singapore Management University, Singapore
More informationLECTURE ON HAC COVARIANCE MATRIX ESTIMATION AND THE KVB APPROACH
LECURE ON HAC COVARIANCE MARIX ESIMAION AND HE KVB APPROACH CHUNG-MING KUAN Institute of Economics Academia Sinica October 20, 2006 ckuan@econ.sinica.edu.tw www.sinica.edu.tw/ ckuan Outline C.-M. Kuan,
More informationEssays on Least Squares Model Averaging
Essays on Least Squares Model Averaging by Tian Xie A thesis submitted to the Department of Economics in conformity with the requirements for the degree of Doctor of Philosophy Queen s University Kingston,
More informationJACKKNIFE MODEL AVERAGING. 1. Introduction
JACKKNIFE MODEL AVERAGING BRUCE E. HANSEN AND JEFFREY S. RACINE Abstract. We consider the problem of obtaining appropriate weights for averaging M approximate (misspecified) models for improved estimation
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot October 28, 2009 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationTime Series and Forecasting Lecture 4 NonLinear Time Series
Time Series and Forecasting Lecture 4 NonLinear Time Series Bruce E. Hansen Summer School in Economics and Econometrics University of Crete July 23-27, 2012 Bruce Hansen (University of Wisconsin) Foundations
More informationMulti-Step Forecast Model Selection
Multi-Step Forecast Model Selection Bruce E. Hansen April 2010 Preliminary Abstract This paper examines model selection and combination in the context of multi-step linear forecasting. We start by investigating
More informationCentral Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Structural Breaks October 29-31, / 91. Bruce E.
Forecasting Lecture 3 Structural Breaks Central Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Structural Breaks October 29-31, 2013 1 / 91 Bruce E. Hansen Organization Detection
More informationOut-of-Sample Return Predictability: a Quantile Combination Approach
Out-of-Sample Return Predictability: a Quantile Combination Approach Luiz Renato Lima a and Fanning Meng a August 8, 2016 Abstract This paper develops a novel forecasting method that minimizes the effects
More informationBootstrapping Heteroskedasticity Consistent Covariance Matrix Estimator
Bootstrapping Heteroskedasticity Consistent Covariance Matrix Estimator by Emmanuel Flachaire Eurequa, University Paris I Panthéon-Sorbonne December 2001 Abstract Recent results of Cribari-Neto and Zarkos
More informationAn estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic
Chapter 6 ESTIMATION OF THE LONG-RUN COVARIANCE MATRIX An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic standard errors for the OLS and linear IV estimators presented
More informationModel averaging, asymptotic risk, and regressor groups
Quantitative Economics 5 2014), 495 530 1759-7331/20140495 Model averaging, asymptotic risk, and regressor groups Bruce E. Hansen University of Wisconsin This paper examines the asymptotic risk of nested
More informationLinear Regression with Time Series Data
Econometrics 2 Linear Regression with Time Series Data Heino Bohn Nielsen 1of21 Outline (1) The linear regression model, identification and estimation. (2) Assumptions and results: (a) Consistency. (b)
More informationThis chapter reviews properties of regression estimators and test statistics based on
Chapter 12 COINTEGRATING AND SPURIOUS REGRESSIONS This chapter reviews properties of regression estimators and test statistics based on the estimators when the regressors and regressant are difference
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot November 2, 2011 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationEcon 423 Lecture Notes: Additional Topics in Time Series 1
Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes
More informationThe Role of "Leads" in the Dynamic Title of Cointegrating Regression Models. Author(s) Hayakawa, Kazuhiko; Kurozumi, Eiji
he Role of "Leads" in the Dynamic itle of Cointegrating Regression Models Author(s) Hayakawa, Kazuhiko; Kurozumi, Eiji Citation Issue 2006-12 Date ype echnical Report ext Version publisher URL http://hdl.handle.net/10086/13599
More informationEcon 582 Nonparametric Regression
Econ 582 Nonparametric Regression Eric Zivot May 28, 2013 Nonparametric Regression Sofarwehaveonlyconsideredlinearregressionmodels = x 0 β + [ x ]=0 [ x = x] =x 0 β = [ x = x] [ x = x] x = β The assume
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
More informationNonstationary Time Series:
Nonstationary Time Series: Unit Roots Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana September
More informationNews Shocks: Different Effects in Boom and Recession?
News Shocks: Different Effects in Boom and Recession? Maria Bolboaca, Sarah Fischer University of Bern Study Center Gerzensee June 7, 5 / Introduction News are defined in the literature as exogenous changes
More informationA near optimal test for structural breaks when forecasting under square error loss
A near optimal test for structural breaks when forecasting under square error loss Tom Boot Andreas Pick December 22, 26 Abstract We propose a near optimal test for structural breaks of unknown timing
More informationVector Autoregressive Model. Vector Autoregressions II. Estimation of Vector Autoregressions II. Estimation of Vector Autoregressions I.
Vector Autoregressive Model Vector Autoregressions II Empirical Macroeconomics - Lect 2 Dr. Ana Beatriz Galvao Queen Mary University of London January 2012 A VAR(p) model of the m 1 vector of time series
More informationJACKKNIFE MODEL AVERAGING. 1. Introduction
JACKKNIFE MODEL AVERAGING BRUCE E. HANSEN AND JEFFREY S. RACINE Abstract. We consider the problem of obtaining appropriate weights for averaging M approximate (misspecified models for improved estimation
More informationA Semiparametric Generalized Ridge Estimator and Link with Model Averaging
A Semiparametric Generalized Ridge Estimator and Link with Model Averaging Aman Ullah, Alan T.K. Wan y, Huansha Wang z, Xinyu hang x, Guohua ou { February 4, 203 Abstract In recent years, the suggestion
More informationECONOMETRICS HONOR S EXAM REVIEW SESSION
ECONOMETRICS HONOR S EXAM REVIEW SESSION Eunice Han ehan@fas.harvard.edu March 26 th, 2013 Harvard University Information 2 Exam: April 3 rd 3-6pm @ Emerson 105 Bring a calculator and extra pens. Notes
More informationVolatility. Gerald P. Dwyer. February Clemson University
Volatility Gerald P. Dwyer Clemson University February 2016 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use
More informationLinear Regression with Time Series Data
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f e c o n o m i c s Econometrics II Linear Regression with Time Series Data Morten Nyboe Tabor u n i v e r s i t y o f c o p e n h a g
More informationTopic 4 Unit Roots. Gerald P. Dwyer. February Clemson University
Topic 4 Unit Roots Gerald P. Dwyer Clemson University February 2016 Outline 1 Unit Roots Introduction Trend and Difference Stationary Autocorrelations of Series That Have Deterministic or Stochastic Trends
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao
More informationGARCH Models. Eduardo Rossi University of Pavia. December Rossi GARCH Financial Econometrics / 50
GARCH Models Eduardo Rossi University of Pavia December 013 Rossi GARCH Financial Econometrics - 013 1 / 50 Outline 1 Stylized Facts ARCH model: definition 3 GARCH model 4 EGARCH 5 Asymmetric Models 6
More informationNote: The primary reference for these notes is Enders (2004). An alternative and more technical treatment can be found in Hamilton (1994).
Chapter 4 Analysis of a Single Time Series Note: The primary reference for these notes is Enders (4). An alternative and more technical treatment can be found in Hamilton (994). Most data used in financial
More informationPIER Working Paper
Penn Institute for Economic Research Department of Economics University of Pennsylvania 378 Locust Walk Philadelphia, PA 904-6297 pier@econ.upenn.edu http://economics.sas.upenn.edu/pier PIER Working Paper
More informationMidterm Suggested Solutions
CUHK Dept. of Economics Spring 2011 ECON 4120 Sung Y. Park Midterm Suggested Solutions Q1 (a) In time series, autocorrelation measures the correlation between y t and its lag y t τ. It is defined as. ρ(τ)
More informationDiscussion of Tests of Equal Predictive Ability with Real-Time Data by T. E. Clark and M.W. McCracken
Discussion of Tests of Equal Predictive Ability with Real-Time Data by T. E. Clark and M.W. McCracken Juri Marcucci Bank of Italy 5 th ECB Workshop on Forecasting Techniques Forecast Uncertainty in Macroeconomics
More informationLASSO-Type Penalties for Covariate Selection and Forecasting in Time Series
Journal of Forecasting, J. Forecast. 35, 592 612 (2016) Published online 21 February 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2403 LASSO-Type Penalties for Covariate Selection
More informationComparing Nested Predictive Regression Models with Persistent Predictors
Comparing Nested Predictive Regression Models with Persistent Predictors Yan Ge y and ae-hwy Lee z November 29, 24 Abstract his paper is an extension of Clark and McCracken (CM 2, 25, 29) and Clark and
More information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
More informationWeighted-Average Least Squares Prediction
Econometric Reviews ISSN: 0747-4938 (Print) 1532-4168 (Online) Journal homepage: http://www.tandfonline.com/loi/lecr20 Weighted-Average Least Squares Prediction Jan R. Magnus, Wendun Wang & Xinyu Zhang
More informationDarmstadt Discussion Papers in Economics
Darmstadt Discussion Papers in Economics The Effect of Linear Time Trends on Cointegration Testing in Single Equations Uwe Hassler Nr. 111 Arbeitspapiere des Instituts für Volkswirtschaftslehre Technische
More informationAugmenting our AR(4) Model of Inflation. The Autoregressive Distributed Lag (ADL) Model
Augmenting our AR(4) Model of Inflation Adding lagged unemployment to our model of inflationary change, we get: Inf t =1.28 (0.31) Inf t 1 (0.39) Inf t 2 +(0.09) Inf t 3 (0.53) (0.09) (0.09) (0.08) (0.08)
More informationECON3327: Financial Econometrics, Spring 2016
ECON3327: Financial Econometrics, Spring 2016 Wooldridge, Introductory Econometrics (5th ed, 2012) Chapter 11: OLS with time series data Stationary and weakly dependent time series The notion of a stationary
More informationDSGE Methods. Estimation of DSGE models: GMM and Indirect Inference. Willi Mutschler, M.Sc.
DSGE Methods Estimation of DSGE models: GMM and Indirect Inference Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@wiwi.uni-muenster.de Summer
More informationQuick Review on Linear Multiple Regression
Quick Review on Linear Multiple Regression Mei-Yuan Chen Department of Finance National Chung Hsing University March 6, 2007 Introduction for Conditional Mean Modeling Suppose random variables Y, X 1,
More informationASSET PRICING MODELS
ASSE PRICING MODELS [1] CAPM (1) Some notation: R it = (gross) return on asset i at time t. R mt = (gross) return on the market portfolio at time t. R ft = return on risk-free asset at time t. X it = R
More informationMonitoring Forecasting Performance
Monitoring Forecasting Performance Identifying when and why return prediction models work Allan Timmermann and Yinchu Zhu University of California, San Diego June 21, 2015 Outline Testing for time-varying
More informationEconometrics Honor s Exam Review Session. Spring 2012 Eunice Han
Econometrics Honor s Exam Review Session Spring 2012 Eunice Han Topics 1. OLS The Assumptions Omitted Variable Bias Conditional Mean Independence Hypothesis Testing and Confidence Intervals Homoskedasticity
More informationThe Bootstrap: Theory and Applications. Biing-Shen Kuo National Chengchi University
The Bootstrap: Theory and Applications Biing-Shen Kuo National Chengchi University Motivation: Poor Asymptotic Approximation Most of statistical inference relies on asymptotic theory. Motivation: Poor
More information9. Model Selection. statistical models. overview of model selection. information criteria. goodness-of-fit measures
FE661 - Statistical Methods for Financial Engineering 9. Model Selection Jitkomut Songsiri statistical models overview of model selection information criteria goodness-of-fit measures 9-1 Statistical models
More informationAveraging Estimators for Regressions with a Possible Structural Break
Averaging Estimators for Regressions with a Possible Structural Break Bruce E. Hansen University of Wisconsin y www.ssc.wisc.edu/~bhansen September 2007 Preliminary Abstract This paper investigates selection
More informationEstimation of Time-invariant Effects in Static Panel Data Models
Estimation of Time-invariant Effects in Static Panel Data Models M. Hashem Pesaran University of Southern California, and Trinity College, Cambridge Qiankun Zhou University of Southern California September
More informationTime-varying sparsity in dynamic regression models
Time-varying sparsity in dynamic regression models Professor Jim Griffin (joint work with Maria Kalli, Canterbury Christ Church University) University of Kent Regression models Often we are interested
More informationReview of Classical Least Squares. James L. Powell Department of Economics University of California, Berkeley
Review of Classical Least Squares James L. Powell Department of Economics University of California, Berkeley The Classical Linear Model The object of least squares regression methods is to model and estimate
More informationEcon 623 Econometrics II Topic 2: Stationary Time Series
1 Introduction Econ 623 Econometrics II Topic 2: Stationary Time Series In the regression model we can model the error term as an autoregression AR(1) process. That is, we can use the past value of the
More informationLinear Instrumental Variables Model Averaging Estimation
Linear Instrumental Variables Model Averaging Estimation Luis F. Martins Department of Quantitative Methods, ISCE-LUI, Portugal Centre for International Macroeconomic Studies CIMS, UK luis.martins@iscte.pt
More informationRegression with time series
Regression with time series Class Notes Manuel Arellano February 22, 2018 1 Classical regression model with time series Model and assumptions The basic assumption is E y t x 1,, x T = E y t x t = x tβ
More informationStatistica Sinica Preprint No: SS R2
Statistica Sinica Preprint No: SS-2017-0034.R2 Title OPTIMAL MODEL AVERAGING OF VARYING COEFFICIENT MODELS Manuscript ID SS-2017-0034.R2 URL http://www.stat.sinica.edu.tw/statistica/ DOI 10.5705/ss.202017.0034
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationUniversity of Pretoria Department of Economics Working Paper Series
University of Pretoria Department of Economics Working Paper Series Predicting Stock Returns and Volatility Using Consumption-Aggregate Wealth Ratios: A Nonlinear Approach Stelios Bekiros IPAG Business
More informationGMM-based Model Averaging
GMM-based Model Averaging Luis F. Martins Department of Quantitative Methods, ISCTE-LUI, Portugal Centre for International Macroeconomic Studies (CIMS), UK (luis.martins@iscte.pt) Vasco J. Gabriel CIMS,
More informationLong-Run Covariability
Long-Run Covariability Ulrich K. Müller and Mark W. Watson Princeton University October 2016 Motivation Study the long-run covariability/relationship between economic variables great ratios, long-run Phillips
More informationLASSO-type penalties for covariate selection and forecasting in time series
LASSO-type penalties for covariate selection and forecasting in time series Evandro Konzen 1 Flavio A. Ziegelmann 2 Abstract This paper studies some forms of LASSO-type penalties in time series to reduce
More informationForecast combination and model averaging using predictive measures. Jana Eklund and Sune Karlsson Stockholm School of Economics
Forecast combination and model averaging using predictive measures Jana Eklund and Sune Karlsson Stockholm School of Economics 1 Introduction Combining forecasts robustifies and improves on individual
More informationEconometric Forecasting
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 1, 2014 Outline Introduction Model-free extrapolation Univariate time-series models Trend
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna November 23, 2013 Outline Introduction
More informationFinancial Econometrics Return Predictability
Financial Econometrics Return Predictability Eric Zivot March 30, 2011 Lecture Outline Market Efficiency The Forms of the Random Walk Hypothesis Testing the Random Walk Hypothesis Reading FMUND, chapter
More informationForecasting the unemployment rate when the forecast loss function is asymmetric. Jing Tian
Forecasting the unemployment rate when the forecast loss function is asymmetric Jing Tian This version: 27 May 2009 Abstract This paper studies forecasts when the forecast loss function is asymmetric,
More informationMultivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8]
1 Multivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8] Insights: Price movements in one market can spread easily and instantly to another market [economic globalization and internet
More informationARIMA Modelling and Forecasting
ARIMA Modelling and Forecasting Economic time series often appear nonstationary, because of trends, seasonal patterns, cycles, etc. However, the differences may appear stationary. Δx t x t x t 1 (first
More informationStatistics 910, #5 1. Regression Methods
Statistics 910, #5 1 Overview Regression Methods 1. Idea: effects of dependence 2. Examples of estimation (in R) 3. Review of regression 4. Comparisons and relative efficiencies Idea Decomposition Well-known
More information1 Motivation for Instrumental Variable (IV) Regression
ECON 370: IV & 2SLS 1 Instrumental Variables Estimation and Two Stage Least Squares Econometric Methods, ECON 370 Let s get back to the thiking in terms of cross sectional (or pooled cross sectional) data
More informationTime Series Econometrics For the 21st Century
Time Series Econometrics For the 21st Century by Bruce E. Hansen Department of Economics University of Wisconsin January 2017 Bruce Hansen (University of Wisconsin) Time Series Econometrics January 2017
More informationTests of Equal Predictive Ability with Real-Time Data
Tests of Equal Predictive Ability with Real-Time Data Todd E. Clark Federal Reserve Bank of Kansas City Michael W. McCracken Board of Governors of the Federal Reserve System April 2007 (preliminary) Abstract
More informationSimple Linear Regression
Simple Linear Regression Christopher Ting Christopher Ting : christophert@smu.edu.sg : 688 0364 : LKCSB 5036 January 7, 017 Web Site: http://www.mysmu.edu/faculty/christophert/ Christopher Ting QF 30 Week
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 17, 2012 Outline Heteroskedasticity
More informationForecasting Levels of log Variables in Vector Autoregressions
September 24, 200 Forecasting Levels of log Variables in Vector Autoregressions Gunnar Bårdsen Department of Economics, Dragvoll, NTNU, N-749 Trondheim, NORWAY email: gunnar.bardsen@svt.ntnu.no Helmut
More informationLinear Regression with Time Series Data
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f e c o n o m i c s Econometrics II Linear Regression with Time Series Data Morten Nyboe Tabor u n i v e r s i t y o f c o p e n h a g
More informationEconomic Forecasting with Many Predictors
University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2017 Economic Forecasting with Many Predictors Fanning Meng University of Tennessee,
More informationEVALUATING DIRECT MULTI-STEP FORECASTS
EVALUATING DIRECT MULTI-STEP FORECASTS Todd Clark and Michael McCracken Revised: April 2005 (First Version December 2001) RWP 01-14 Research Division Federal Reserve Bank of Kansas City Todd E. Clark is
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationLeast Squares Estimation-Finite-Sample Properties
Least Squares Estimation-Finite-Sample Properties Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Finite-Sample 1 / 29 Terminology and Assumptions 1 Terminology and Assumptions
More informationTitle. Description. Quick start. Menu. stata.com. xtcointtest Panel-data cointegration tests
Title stata.com xtcointtest Panel-data cointegration tests Description Quick start Menu Syntax Options Remarks and examples Stored results Methods and formulas References Also see Description xtcointtest
More informationCointegrated VAR s. Eduardo Rossi University of Pavia. November Rossi Cointegrated VAR s Financial Econometrics / 56
Cointegrated VAR s Eduardo Rossi University of Pavia November 2013 Rossi Cointegrated VAR s Financial Econometrics - 2013 1 / 56 VAR y t = (y 1t,..., y nt ) is (n 1) vector. y t VAR(p): Φ(L)y t = ɛ t The
More informationPanel Threshold Regression Models with Endogenous Threshold Variables
Panel Threshold Regression Models with Endogenous Threshold Variables Chien-Ho Wang National Taipei University Eric S. Lin National Tsing Hua University This Version: June 29, 2010 Abstract This paper
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationDYNAMIC ECONOMETRIC MODELS Vol. 9 Nicolaus Copernicus University Toruń Mariola Piłatowska Nicolaus Copernicus University in Toruń
DYNAMIC ECONOMETRIC MODELS Vol. 9 Nicolaus Copernicus University Toruń 2009 Mariola Piłatowska Nicolaus Copernicus University in Toruń Combined Forecasts Using the Akaike Weights A b s t r a c t. The focus
More informationTesting methodology. It often the case that we try to determine the form of the model on the basis of data
Testing methodology It often the case that we try to determine the form of the model on the basis of data The simplest case: we try to determine the set of explanatory variables in the model Testing for
More informationBagging and Forecasting in Nonlinear Dynamic Models
DBJ Discussion Paper Series, No.0905 Bagging and Forecasting in Nonlinear Dynamic Models Mari Sakudo (Research Institute of Capital Formation, Development Bank of Japan, and Department of Economics, Sophia
More informationSupplemental Material for KERNEL-BASED INFERENCE IN TIME-VARYING COEFFICIENT COINTEGRATING REGRESSION. September 2017
Supplemental Material for KERNEL-BASED INFERENCE IN TIME-VARYING COEFFICIENT COINTEGRATING REGRESSION By Degui Li, Peter C. B. Phillips, and Jiti Gao September 017 COWLES FOUNDATION DISCUSSION PAPER NO.
More information