Markov-switching MIDAS models Pierre Gu erin Massimiliano - PowerPoint PPT Presentation

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Markov-switching MIDAS models Pierre Gu´ erin Massimiliano Marcellino European University Institute September 29, 2010 Eurostat 6th Colloquium on Modern Tools for Business Cycle Analysis

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Introduction We propose a new model - Markov-switching MIDAS (MS-MIDAS) - that proves to be useful for short-term forecasts of quarterly GDP. Our model includes: mixed-frequency data non-linearities Our model is relatively simple to use and also produces real time probabilities of recession.

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Outline Markov-switching MIDAS models 1 Monte Carlo experiments 2 Applications to the prediction of quarterly GDP 3

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP MIDAS models MIDAS models were introduced by Ghysels, Santa-Clara and Valkanov (2004). y t = β 0 + β 1 B ( L 1 / m ; θ ) x m t − h + ǫ t where B ( L 1 / m ; θ ) = � K j =1 b ( j ; θ ) L ( j − 1) / m The weighting function b ( j ; θ ) is defined as: exp ( θ 1 j + ... + θ Q j Q ) b ( j ; θ ) = � K j =1 exp ( θ 1 j + ... + θ Q j Q ) Autoregressive dynamics has to be introduced through a common factor: y t = β 0 + λ y t − d + β 1 B ( L 1 / m ; θ )(1 − λ L d ) x ( m ) t − h + ǫ t

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Markov-switching MIDAS models We incorporate regime changes to the AR-MIDAS using Markov-switching process: y t = β 0 ( S t )+ λ y t − d + β 1 ( S t ) B ( L (1 / m ) ; θ )(1 − λ L d ) x ( m ) t − h + ǫ t ( S t ) where ǫ t | S t ∼ NID (0 , σ 2 ( S t )) S t is a Markov-chain with M regimes defined by the following transition probabilities: p ij = Pr ( S t +1 = j | S t = i ) M � p ij = 1 ∀ i , j ǫ { 1 , ..., M } j =1 Estimation is carried out with Maximum Likelihood.

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: D.G.P. We consider a model with two regimes, and switches in the parameters β 0 , β 1 and in the variance σ 2 (MSH(2)-MIDAS model). We consider the following true parameter values for the parameters: ( β 0 , 1 , β 0 , 2 ) = ( − 1 , 1) , ( β 1 , 1 , β 1 , 2 ) = (0 . 6 , 0 . 2) , λ = 0 . 2 ( σ 1 , σ 2 ) = (1 , 0 . 67) , ( θ 1 , θ 2 ) = (2 ∗ 10 − 1 , − 3 ∗ 10 − 2) We use two sets of parameters for the transition probabilities: ( p 11 , p 22 ) = (0 . 95 , 0 . 95) and ( p 11 , p 22 ) = (0 . 85 , 0 . 95)

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: In-sample exercise The dependent variable is constructed using the outcome of a Markov-chain, simulated series for x t , and the true parameter values. We repeat the experiment 1000 times and report the means of the maximum likelihood estimates and the standard deviations. We use an approximation error for assessing the accuracy of the estimation for the MIDAS parameters ( θ 1 , θ 2 ): � M = K j =1 [ b ( j , ˆ θ ) − b ( j , θ )] 2 � M = K b ( j , θ ) 2 j =1

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: In-sample results (I) p 11 =0.95 p 22 =0.95 β 0 , 1 =-1 β 0 , 2 =1 β 1 , 1 =0.6 β 1 , 2 =0.2 λ = 0 . 2 Approx. error K=3 T=200 0.936 0.946 -1.043 1.042 0.504 0.170 0.165 0.262 (0.039) (0.032) (0.224) (0.140) (0.103) (0.056) (0.074) T=500 0.944 0.949 -1.042 1.028 0.532 0.181 0.174 0.157 (0.018) (0.016) (0.116) (0.074) (0.045) (0.023) (0.042) K=13 T=200 0.935 0.945 -1.047 1.041 0.501 0.174 0.163 0.256 (0.040) (0.028) (0.242) (0.137) (0.104) (0.046) (0.074) T=500 0.944 0.949 -1.036 1.037 0.532 0.181 0.172 0.174 (0.019) (0.016) (0.117) (0.077) (0.044) (0.022) (0.044)

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: In-sample results (II) p 11 =0.85 p 22 =0.95 β 0 , 1 =-1 β 0 , 2 =1 β 1 , 1 =0.6 β 1 , 2 =0.2 λ = 0 . 2 Approx. error K=3 T=200 0.808 0.951 -1.015 1.030 0.455 0.176 0.178 0.237 (0.114) (0.025) (0.455) (0.116) (0.211) (0.033) (0.069) T=500 0.832 0.952 -1.025 1.026 0.506 0.180 0.181 0.133 (0.053) (0.014) (0.185) (0.065) (0.081) (0.017) (0.040) K=13 T=200 0.816 0.950 -0.993 1.035 0.451 0.175 0.176 0.234 (0.097) (0.028) (0.423) (0.121) (0.193) (0.032) (0.070) T=500 0.833 0.951 -1.032 1.022 0.507 0.181 0.185 0.149 (0.051) (0.014) (0.191) (0.065) (0.076) (0.017) (0.041)

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: Forecasting exercise (I) We use the same D.G.P. as in the in-sample exercise (with 13 lags for the high frequency indicator). We use seven different models to compute the forecasts: the MSHAR(2)-MIDAS model (i.e. the true model), the MSH(2)-MIDAS model (i.e. the true model without an autoregressive lag), a standard MIDAS and AR-MIDAS models. We also use an AR(1) model and a standard Markov-switching model with two regimes (MSIHAR(2) model) as benckmarks The sample size T is split between an estimation sample and an evaluation sample We use three different sizes for the evaluation sample (20, 50, 100) and compute one-step ahead forecasts by recursively expanding the estimation sample.

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: Forecasting exercise (II) We use three different criteria for comparing the forecasting performance: the MSFE, Quadratic Probability Score (QPS) and Log Probability Score (LPS). QPS and LPS are defined as: T QPS = 2 � ( P ( S t +1 = 1) − S t +1 ) 2 T t =1 T LPS = − 1 � (1 − S t +1 ) log (1 − P ( S t +1 = 1)) + S t +1 log ( P ( S t +1 = 1)) T t =1 We repeat the experiment 200 times and take the average of the MSFE, QPS and LPS over the number of replications.

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: Forecasting Results (I) Number of out-sample forecasts H : 20 50 100 Sample Size T=200 QPS LPS MSFE QPS LPS MSFE QPS LPS MSFE MSHAR(2)-MIDAS 0.398 0.609 1.432 0.300 0.447 1.782 0.295 0.464 1.207 MSIHAR(2) 0.392 0.642 1.709 0.370 0.584 1.617 0.872 1.690 1.301 MSH(2)-MIDAS 0.412 0.635 1.468 0.258 0.398 1.860 0.304 0.469 1.226 MSIHAR(2)-MIDAS 0.444 0.683 1.646 0.356 0.561 1.671 0.821 1.629 1.237 AR-MIDAS - - 1.563 - - 1.722 - - 1.423 MIDAS - - 2.510 - - 2.432 - - 1.738 AR(1) - - 1.596 - - 1.563 - - 1.456

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Monte carlo experiments: Forecasting Results (II) Number of out-sample forecasts H : 20 50 100 Sample Size T=500 QPS LPS MSFE QPS LPS MSFE QPS LPS MSFE MSHAR(2)-MIDAS 0.456 0.653 1.385 0.238 0.404 1.487 0.421 0.618 1.768 MSIHAR(2) 0.358 0.543 1.787 0.365 0.547 2.034 0.405 0.600 2.066 MSH(2)-MIDAS 0.524 0.736 1.442 0.216 0.374 1.519 0.419 0.622 1.755 MSIHAR(2)-MIDAS 0.361 0.546 1.326 0.326 0.498 1.625 0.413 0.608 1.888 AR-MIDAS - - 1.410 - - 1.711 - - 1.898 MIDAS - - 1.893 - - 2.202 - - 2.451 AR(1) - - 1.832 - - 1.931 - - 2.032

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP Prediction of the US GDP We use data for GDP from the Real-time database of the Federal Reserve Bank of Philadelphia ( t goes from 1959:Q1 to 2009:Q4). We consider the slope of the yield curve, the S&P 500 index and the Federal Funds as potential predictors for quarterly GDP. We use information criteria for selecting the number of regimes, and the parametrization of the model. The model that gets the best fit is the one with three regimes, and switches in β 0 , β 1 and in the variance σ 2 .

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP In-sample Smoothed Probabilities Figure 1: MS-MIDAS Quarterly GDP and monthly slope of the yield curve Sample 1959:Q1 - 2009:Q4 Panel A: Smoothed Probabilities of being in a recession 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 Panel B: Smoothed Probabilities of being in the low expansion regime 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 Panel C: Smoothed Probabilities of being in the high expansion regime 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007