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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

1

Markov-switching MIDAS models

2

Monte Carlo experiments

3

Applications to the prediction of quarterly GDP

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). yt = β0 + β1B(L1/m; θ)xm

t−h + ǫt

where B(L1/m; θ) = K

j=1 b(j; θ)L(j−1)/m

The weighting function b(j; θ) is defined as: b(j; θ) = exp(θ1j + ... + θQjQ) K

j=1 exp(θ1j + ... + θQjQ)

Autoregressive dynamics has to be introduced through a common factor: yt = β0 + λyt−d + β1B(L1/m; θ)(1 − λLd)x(m)

t−h + ǫt

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: yt = β0(St)+λyt−d +β1(St)B(L(1/m); θ)(1−λLd)x(m)

t−h+ǫt(St)

where ǫt|St ∼ NID(0, σ2(St)) St is a Markov-chain with M regimes defined by the following transition probabilities: pij = Pr(St+1 = j|St = i)

M

• j=1

pij = 1∀i, jǫ{1, ..., M} 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 β0, β1 and in the variance σ2 (MSH(2)-MIDAS model). We consider the following true parameter values for the parameters: (β0,1, β0,2) = (−1, 1), (β1,1, β1,2) = (0.6, 0.2), λ = 0.2 (σ1, σ2) = (1, 0.67), (θ1, θ2) = (2 ∗ 10−1, −3 ∗ 10−2) We use two sets of parameters for the transition probabilities: (p11, p22) = (0.95, 0.95)and(p11, p22) = (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 xt, 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 (θ1, θ2): M=K

j=1 [b(j, ˆ

θ) − b(j, θ)]2 M=K

j=1

b(j, θ)2

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Monte carlo experiments: In-sample results (I)

p11=0.95 p22=0.95 β0,1=-1 β0,2=1 β1,1=0.6 β1,2=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)

p11=0.85 p22=0.95 β0,1=-1 β0,2=1 β1,1=0.6 β1,2=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: QPS = 2 T

T

• t=1

(P(St+1 = 1) − St+1)2 LPS = − 1 T

T

• t=1

(1 − St+1)log(1 − P(St+1 = 1)) + St+1log(P(St+1 = 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 β0, β1 and in the variance σ2.

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

In-sample Smoothed Probabilities

Sample 1959:Q1 - 2009:Q4 Figure 1: MS-MIDAS Quarterly GDP and monthly slope of the yield curve Panel A: Smoothed Probabilities of being in a recession

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 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

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 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

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

In-sample Results: Estimated weights

Figure 2: Weights of the AR-MIDAS (LHS) and MSHAR(3)-MIDAS (RHS) exponential lag polynomial Slope of the yield curve 0,05 0,1 0,15 0,2 0,25 1 2 3 4 5 6 7 8 9 10 11 12 S&P 500 0,02 0,04 0,06 0,08 0,1 0,12 0,14 1 2 3 4 5 6 7 8 9 10 11 12 Fed Funds 0,05 0,1 0,15 0,2 0,25 1 2 3 4 5 6 7 8 9 10 11 12 Slope of the yield curve 0,05 0,1 0,15 0,2 1 2 3 4 5 6 7 8 9 10 11 12 S&P 500 0,02 0,04 0,06 0,08 0,1 0,12 0,14 1 2 3 4 5 6 7 8 9 10 11 12 Fed Funds 0,05 0,1 0,15 0,2 0,25 0,3 0,35 1 2 3 4 5 6 7 8 9 10 11 12

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Design of the Out-of-sample forecasts

The evaluation sample goes from t = 1998 : Q1 to 2009 : Q4 Forecasts are computed using the direct method and by recursively expanding the sample size We compute forecasts with horizon h = {0, 1/3, 2/3, 1, 4/3, 5/3, 2} for yt+1 and yt+2 for the MIDAS, AR-MIDAS, MSIH-MIDAS, MSIHAR-MIDAS and MSIHAR models. For the MS models, we also need to compute forecasts for the filtered probabilities, this is done recursively: P(St+k = j|xt) =

M

• i=1

pijP(St+k−1 = i|xt)

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Out-of-sample results: Relative MSFE

Forecast horizon (h) Model 1/3 2/3 1 4/3 5/3 2 Slope of the MSIH(3)-MIDAS 1.080 1.061 1.066 1.066 1.022 1.002 1.009 yield curve MSIHAR(3)-MIDAS 0.960 1.019 1.038 0.964 1.032 1.049 1.041 MSH(3)-MIDAS 1.080 1.043 1.059 1.025 1.067 1.073 1.049 MSHAR(3)-MIDAS 1.073 1.073 1.044 0.992 0.996 1.019 1.015 AR-MIDAS 1.037 1.022 1.022 1.010 0.995 0.997 0.997 MIDAS 1.242 1.231 1.231 1.252 1.065 1.069 1.276 S&P 500 MSIH(3)-MIDAS 0.691 0.739 0.712 0.765 0.740 0.784 0.785 MSIHAR(3)-MIDAS 0.679 0.690 0.639 0.726 0.775 0.776 0.792 MSH(3)-MIDAS 0.913 0.876 0.904 1.078 0.872 0.902 0.861 MSHAR(3)-MIDAS 0.903 1.160 0.932 1.128 0.873 0.885 0.845 AR-MIDAS 0.769 0.726 0.715 0.746 0.715 0.754 0.779 MIDAS 0.762 0.728 0.713 0.729 0.684 0.727 0.767 Fed Funds MSIH(3)-MIDAS 0.909 0.996 0.983 1.005 1.074 1.086 1.284 MSIHAR(3)-MIDAS 0.944 0.952 0.995 1.044 1.009 1.046 1.185 MSH(3)-MIDAS 1.041 0.916 0.963 0.997 1.126 1.110 1.119 MSHAR(3)-MIDAS 0.974 0.886 0.983 0.971 0.966 1.172 1.009 AR-MIDAS 0.874 0.942 0.945 0.945 1.081 1.124 1.278 MIDAS 1.041 1.078 1.171 1.174 1.237 1.237 1.442 MSIHAR(3)

• 1.071
• 0.984

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Out-of-sample results: Quadratic Probability Score

Forecast horizon (h) Model 1/3 2/3 1 4/3 5/3 2 Slope of the MSIH(3)-MIDAS 0.391 0.395 0.395 0.395 0.436 0.448 0.469 yield curve MSIHAR(3)-MIDAS 0.384 0.374 0.374 0.383 0.433 0.481 0.468 MSH(3)-MIDAS 0.270 0.296 0.297 0.407 0.485 0.461 0.402 MSHAR(3)-MIDAS 0.270 0.287 0.306 0.370 0.448 0.494 0.491 S&P 500 MSIH(3)-MIDAS 0.399 0.449 0.469 0.439 0.465 0.437 0.457 MSIHAR(3)-MIDAS 0.434 0.436 0.443 0.408 0.520 0.519 0.509 MSH(3)-MIDAS 0.442 0.414 0.382 0.412 0.473 0.495 0.517 MSHAR(3)-MIDAS 0.391 0.287 0.410 0.353 0.456 0.466 0.635 Fed Funds MSIH(3)-MIDAS 0.391 0.375 0.388 0.391 0.385 0.390 0.463 MSIHAR(3)-MIDAS 0.394 0.401 0.400 0.385 0.459 0.462 0.435 MSH(3)-MIDAS 0.415 0.369 0.374 0.337 0.483 0.389 0.495 MSHAR(3)-MIDAS 0.423 0.403 0.436 0.411 0.489 0.502 0.445 MSIHAR(3)

• 0.357
• 0.452

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Out-of-sample results: Estimated Probabilities of recession

Figure 3: Nowcasted Probabilities of recession, Real-time data, 1997:Q4-2010:Q2, MSH(3)-MIDAS model with the monthly slope of the yield curve, forecast horizon h=0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1997 1998 1998 1999 1999 2000 2000 2001 2001 2002 2002 2003 2003 2004 2004 2005 2005 2006 2006 2007 2007 2008 2008 2009 2009 2010

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Probabilities of recession updated on a monthly basis

Quarter Month P(St = 1) NBERT Quarter Month P(St = 1) NBERT January 0.146 January 0.995 1 Q1 2007 February 0.137 Q1 2009 February 1.000 1 March 0.148 March 0.999 1 April 0.128 April 0.994 1 Q2 2007 May 0.123 Q2 2009 May 0.987 1 June 0.127 June 0.995 1 July 0.053 July 0.829 Q3 2007 August 0.036 Q3 2009 August 0.822 September 0.033 September 0.824 October 0.283 October 0.663 Q4 2007 November 0.241 Q4 2009 November 0.639 December 0.268 1 December 0.642 January 0.636 1 January 0.486 Q1 2008 February 0.742 1 Q1 2010 February 0.468 March 0.878 1 March 0.479 April 0.931 1 April 0.339 Q2 2008 May 0.940 1 Q2 2010 May 0.343 June 0.961 1 June 0.345 July 0.985 1 Q3 2008 August 0.983 1 September 0.987 1 October 1.000 1 Q4 2008 November 1.000 1 December 1.000 1

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Prediction of the UK GDP

We use data for GDP from the Bank of England Real-time database (t goes from 1975:Q1 to 2010:Q1). We consider the slope of the yield curve, the FT All Shares Index and the Bank of England Real-Time Database as potential predictors for quarterly GDP. We select a model with two regimes, constant variance and switches in β0 and β1. The evaluation sample goes from t=2004:Q1 to 2010:Q1.

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

In-sample Smoothed Probabilities

Figure 4: MSIAR(2)-MIDAS Quarterly GDP and monthly slope of the yield curve Sample 1975:Q1 - 2010:Q1 Panel A: Smoothed probabilities of being in a recession 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 Panel B: Smoothed Probabilities of being in an expansion 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Out-of-sample results: Relative MSFE

Forecast horizon (h) Model 1/3 2/3 1 4/3 5/3 2 Slope of the MSI(2)-MIDAS 0.621 0.714 0.617 0.693 1.088 1.151 1.130 yield curve MSIAR(2)-MIDAS 0.684 0.684 0.681 0.869 1.069 1.063 1.167 MS(2)-MIDAS 0.801 0.896 0.819 0.844 1.070 1.168 1.164 MSAR(2)-MIDAS 0.698 0.686 0.684 0.802 1.097 1.222 1.133 AR-MIDAS 0.857 0.917 0.880 0.939 0.954 0.953 0.975 MIDAS 0.906 0.969 0.897 1.017 1.115 1.174 1.165 FT All Shares MSI(2)-MIDAS 0.697 0.699 0.861 0.704 1.019 1.009 1.006 MSIAR(2)-MIDAS 0.814 0.810 0.810 0.806 1.070 0.998 0.994 MS(2)-MIDAS 0.858 0.790 1.024 1.201 1.113 1.129 1.129 MSAR(2)-MIDAS 0.750 0.801 0.916 0.766 1.190 1.041 1.040 AR-MIDAS 0.868 0.868 0.859 0.933 0.958 0.939 0.938 MIDAS 0.957 0.897 0.899 0.887 0.956 0.925 0.921 BoE base rate MSI(2)-MIDAS 0.633 0.685 0.828 0.752 1.099 1.095 1.137 MSIAR(2)-MIDAS 0.746 0.788 0.833 0.859 1.088 1.070 1.105 MS(2)-MIDAS 0.637 0.767 0.698 0.698 0.925 0.925 1.149 MSH(2)-MIDAS 0.792 0.757 0.833 0.843 0.917 0.920 1.113 AR-MIDAS 0.953 0.957 0.954 1.078 0.862 0.903 1.079 MIDAS 1.034 1.028 1.032 1.089 1.131 1.132 1.280 MSI(2)

• 0.866
• 1.082

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Out-of-sample results: Quadratic Probability Score

Forecast horizon (h) Model 1/3 2/3 1 4/3 5/3 2 Slope of the MSI(2)-MIDAS 0.204 0.215 0.203 0.201 0.434 0.379 0.345 yield curve MSIAR(2)-MIDAS 0.174 0.174 0.174 0.184 0.403 0.329 0.447 MS(2)-MIDAS 0.227 0.232 0.237 0.234 0.354 0.383 0.323 MSAR(2)-MIDAS 0.182 0.173 0.173 0.178 0.356 0.341 0.391 Share prices MSI(2)-MIDAS 0.263 0.259 0.244 0.237 0.320 0.333 0.345 MSIAR(2)-MIDAS 0.206 0.206 0.189 0.192 0.398 0.411 0.410 MS(2)-MIDAS 0.260 0.196 0.321 0.180 0.383 0.388 0.389 MSAR(2)-MIDAS 0.367 0.469 0.326 0.432 0.488 0.474 0.474 BoE base rate MSI(2)-MIDAS 0.214 0.215 0.213 0.187 0.332 0.339 0.328 MSIAR(2)-MIDAS 0.176 0.177 0.176 0.170 0.371 0.337 0.287 MS(2)-MIDAS 0.154 0.209 0.175 0.175 0.239 0.239 0.335 MSAR(2)-MIDAS 0.267 0.174 0.167 0.175 0.316 0.250 0.346 MSIAR(2)

• 0.175
• 0.315

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Probabilities of recession updated on a monthly basis

Quarter Month P(St = 1) ECRIT Quarter Month P(St = 1) ECRIT January 0.001 January 1.000 1 Q1 2007 February 0.001 Q1 2009 February 1.000 1 March 0.001 March 1.000 1 April 0.001 April 1.000 1 Q2 2007 May 0.001 Q2 2009 May 0.999 1 June 0.001 June 1.000 1 July 0.000 July 0.993 1 Q3 2007 August 0.000 Q3 2009 August 0.991 1 September 0.000 September 0.991 1 October 0.001 October 0.913 1 Q4 2007 November 0.001 Q4 2009 November 0.892 1 December 0.001 December 0.889 1 January 0.004 January 0.523 1 Q1 2008 February 0.004 Q1 2010 February 0.437 1 March 0.004 March 0.390 1 April 0.019 Q2 2008 May 0.019 1 June 0.019 1 July 0.683 1 Q3 2008 August 0.691 1 September 0.683 1 October 1.000 1 Q4 2008 November 1.000 1 December 1.000 1

Markov-switching MIDAS models Monte Carlo experiments Applications to the prediction of quarterly GDP

Conclusions

The MS-MIDAS can improve on the forecasting performance

• f the MIDAS and standard Markov-switching models.

The MS-MIDAS provides relevant information about the state

• f the economy.