Performance analysis of nowcasting of GDP growth when allowing for conditional heteroscedasticity and non-Gaussianity

Figures & data

Figure 1. GDP growth series.

Note: Percent on vertical axes.

Figure 2. Histograms of GDP growth rates.

Note: Number of observations in each bin on vertical axes. Percent on horizontal axes.

Table 1. Descriptive statistics and Jarque-Bera test statistic

Country	Mean	Variance	Skewness	Kurtosis	Jarque-Bera
Australia	0.771	0.517	0.035	4.047	7.40
Canada	0.586	0.538	−0.697	4.190	22.53
France	0.446	0.214	−0.791	6.284	89.12
Japan	0.473	0.978	−0.918	7.455	152.82
United Kingdom	0.515	0.494	−1.112	6.540	117.25
United States	0.664	0.492	−0.867	5.561	64.15

Note: The critical value at the five percent level of the Jarque-Bera test is 5.99.

Table 2. In-sample estimation results

		Australia				Canada
		(1)	(2)	(3)	(4)	(1)	(2)	(3)	(4)
Mean equation [AR(1)]	${\gamma }_0$	0.603	0.604	0.619	0.620	0.291	0.280	0.349	0.348
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	${\gamma }_1$	0.218	0.196	0.121	0.119	0.502	0.536	0.403	0.425
		(0.001)	(0.002)	(0.118)	(0.117)	(0.000)	(0.000)	(0.000)	(0.000)
GARCH equation	$\omega$			<0.000	<0.000			0.130	0.113
				(0.999)	(0.999)			(0.007)	(0.088)
	$\alpha$			0.059	0.058			0.364	0.358
				(0.076)	(0.114)			(0.019)	(0.040)
	$\beta$			0.932	0.932			0.330	0.384
				(0.000)	(0.000)			(0.096)	(0.136)
Degrees of freedom	$\kappa$		5.611		22.050		6.291		12.181
			[3.354]		[37.675]		[3.319]		[10.604]
ARCH-test		10.973	11.212	1.103	1.097	6.834	5.906	0.671	0.918
		(0.027)	(0.024)	(0.894)	(0.895)	(0.145)	(0.206)	(0.955)	(0.922)
JB-test		6.708	0.421^a	0.534	0.002^a	13.684	0.470^a	7.354	2.166^a
		(0.035)	(>0.500)	(>0.500)	(>0.500)	(0.007)	(>0.500)	(0.029)	(0.339)
T		161	161	161	161	161	161	161	161
		France				Japan*
		(1)	(2)	(3)	(4)	(1)	(2)	(3)	(4)
Mean equation [AR(1)]	${\gamma }_0$	0.207	0.248	0.230	0.241	0.386	0.430	0.402	0.405
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	${\gamma }_1$	0.536	0.474	0.492	0.478	0.181	0.134	0.160	0.162
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.025)	(0.061)	(0.066)
GARCH equation	$\omega$			0.051	0.064			0.438	0.440
				(0.290)	(0.417)			(0.001)	(0.002)
	$\alpha$			0.173	0.128			0.546	0.541
				(0.150)	(0.280)			(0.001)	(0.003)
	$\beta$			0.496	0.442
				(0.193)	(0.427)
Degrees of freedom	$\kappa$		6.687		9.916		4.856		38.519
			[3.790]		[8.688]		[1.839]		[137.340]
ARCH-test		8.926	9.859	7.120	6.619	27.785	23.724	2.326	2.246
		(0.063)	(0.042)	(0.130)	(0.157)	(0.000)	(0.000)	(0.676)	(0.691)
JB-test		14.102	2.284^a	4.505	1.186^a	75.542	1.988^a	1.095	0.786^a
		(0.007)	(0.319)	(0.078)	(>0.500)	(0.000)	(0.370)	(>0.500)	(>0.500)
T		161	161	161	161	158	158	158	158
		United Kingdom				United States
		(1)	(2)	(3)	(4)	(1)	(2)	(3)	(4)
Mean equation [AR(1)]	${\gamma }_0$	0.251	0.362	0.254	0.336	0.406	0.481	0.429	0.481
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	${\gamma }_1$	0.507	0.343	0.551	0.421	0.388	0.335	0.403	0.329
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
GARCH equation	$\omega$			0.047	0.070			0.054	0.049
				(0.017)	(0.144)			(0.191)	(0.267)
	$\alpha$			0.213	0.402			0.153	0.210
				(0.011)	(0.146)			(0.004)	(0.086)
	$\beta$			0.642	0.510			0.700	0.684
				(0.000)	(0.003)			(0.000)	(0.000)
Degrees of freedom	$\kappa$		2.048		3.520		3.661		4.987
			[0.617]		[0.276]		[1.390]		[2.107)]
ARCH-test		14.134	21.041	1.274	1.085	31.558	26.806	4.278	2.386
		(0.007)	(0.000)	(0.866)	(0.897)	(0.000)	(0.000)	(0.370)	(0.665)
JB-test		24.809	38.338^a	23.198	2.766^a	51.019	4.808^a	56.434	3.054^a
		(0.002)	(0.000)	(0.002)	(0.251)	(0.000)	(0.090)	(0.000)	(0.217)
T		161	161	161	161	161	161	161	161

Note: The table presents estimation results of the univariate GDP nowcasting model. Model (1) assumes a conditionally homoscedastic standard normal distribution. Model (2) assumes a conditionally homoscedastic t-distribution. Model (3) allows for GARCH(1,1) error terms with standard normal innovations. Model (4) allows for GARCH(1,1) error terms with t-distributed innovations. Standard errors are calculated using the outer product gradient method. The ARCH test is Engle’s test for conditional heteroscedasticity (Engle 1982) using four lags, and the JB-test is the Jarque-Bera test for normality (Jarque and Bera 1980). “a” indicates that the Jarque-Bera test is performed on the probability integral transform of the (standardized) residuals (see footnote 17). T is the number of observations. p-values are in parentheses (). Standard errors for the degrees of freedom in brackets []^{. *} For Japan, the lower bound ($\beta =0$) becomes a binding constraint in the maximum likelihood estimation and no estimate for $\beta$ is accordingly reported.

Table 3. Nowcast evaluation results for GDP growth

	Australia					Canada
Error	RMSE	DM	KL	KS	AD	RMSE	DM	KL	KS	AD
Normal	0.574	-	0.131	0.143	4.439	0.526	-	0.067	0.117	2.864
t	0.573	2.573	0.131	0.142	4.433	0.528	−0.230	0.094	0.108	2.245
GARCH-Normal	0.567	2.147	0.078	0.114	1.782	0.526	0.101	0.010	0.074	0.889
GARCH-t	0.567	2.177	0.078	0.114	1.738	0.526	0.173	0.015	0.074	0.913
	France					Japan
	RMSE	DM	KL	KS	AD	RMSE	DM	KL	KS	AD
Normal	0.378	-	0.048	0.105	1.277	1.032	-	0.133	0.213	7.004
t	0.378	0.126	0.023	0.119	1.360	1.011	1.034	0.152	0.214	7.413
GARCH-Normal	0.376	0.445	0.054	0.111	1.372	0.990	1.822	0.126	0.188	5.772
GARCH-t	0.377	0.321	0.044	0.110	1.400	0.990	1.803	0.134	0.188	5.797
	United Kingdom					United States
	RMSE	DM	KL	KS	AD	RMSE	DM	KL	KS	AD
Normal	0.466	-	0.267	0.168	5.953	0.552	-	0.111	0.150	3.375
t	0.481	−0.824	0.273	0.177	7.622	0.562	−1.343	0.065	0.151	2.942
GARCH-Normal	0.467	−0.192	0.081	0.112	1.449	0.561	−2.181	0.077	0.143	2.282
GARCH-t	0.479	−1.234	0.073	0.119	1.360	0.564	−1.493	0.066	0.150	2.585

Note: The table shows nowcasting performance evaluation results. The first column contains the root mean squared error of the point nowcasts, and the second column presents the Diebold-Mariano test statistic where the model with homoscedastic, normally distributed errors is chosen to be the benchmark. The third column is the Kullback-Leibler divergence. The fourth column gives the Kolmogorov–Smirnov test statistic and the fifth column the Anderson–Darling test statistic. All three density nowcast measures are based on the probability integral transform of the density nowcasts produced by each model. The results are based on N=100 out-of-sample periods and an expanding window for parameter estimation. The critical value at the five percent level for the Kolmogorov–Smirnov test is 0.136 for N=100 and for the Anderson–Darling test it is 2.492.

Figure A1. Conditional standard deviation of the innovation based on the AR-GARCH model.

Note: The plot shows the filtered conditional volatility of the output growth series using Gaussian (solid line) and t-distributed (dashed line) innovations. Percent on vertical axes.

Figure A2. Probability integral transform – Australia.

Note: Vertical axes give relative frequency.

Figure A3. Probability integral transform – Canada.

Note: Vertical axes give relative frequency.

Figure A4. Probability integral transform – France.

Note: Vertical axes give relative frequency.

Figure A5. Probability integral transform – Japan.

Note: Vertical axes give relative frequency.

Figure A6. Probability integral transform – United Kingdom.

Note: Vertical axes give relative frequency.

Figure A7. Probability integral transform – United States.

Note: Vertical axes give relative frequency

Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (4): 987–1007. doi:10.2307/1912773.

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Jarque, C. M., and A. K. Bera. 1980. “Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals.” Economics Letters 6 (3): 255–259. doi:10.1016/0165-1765(80)90024-5.

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