Forecasting a Nonstationary Time Series Using a Mixture of Stationary and Nonstationary Factors as Predictors

Figures & data

Table 1 Coverage rates (%) of 95% prediction intervals for one-step ahead forecasts.

		$\rho =0.0$				$\rho =0.5$				$\rho =0.9$
	T\N	30	50	100	200	30	50	100	200	30	50	100	200
DGP1	30	93	93	91	92	94	94	92	92	94	93	92	91
	50	91	94	93	90	92	94	92	89	92	94	92	90
	100	90	89	91	88	92	90	91	89	91	90	91	88
	200	92	90	89	90	92	90	89	89	93	89	90	89
DGP2	30	97	96	95	94	95	95	95	94	97	98	97	95
	50	96	96	98	94	97	97	94	94	98	97	96	95
	100	97	95	97	95	96	95	95	95	96	96	95	95
	200	96	96	96	94	96	96	94	94	97	96	95	94
DGP3	30	97	97	96	94	97	96	96	95	97	98	95	96
	50	97	95	96	96	97	95	96	96	98	96	96	95
	100	97	96	95	97	96	96	96	95	96	95	96	96
	200	95	97	95	96	95	95	95	95	96	94	95	96

Table 2 Values of $R_{os}^2$ for the performance of mixture-FAR model relative to the corresponding nonstationary-FAR model.

	$\rho =0.0$				$\rho =0.5$				$\rho =0.9$
T1\N	30	50	100	200	30	50	100	200	30	50	100	200
40	0.43	0.44	0.45	0.46	0.41	0.42	0.42	0.43	0.33	0.34	0.35	0.35
60	0.45	0.46	0.46	0.47	0.42	0.43	0.44	0.45	0.36	0.36	0.36	0.37
100	0.46	0.47	0.48	0.48	0.44	0.45	0.46	0.46	0.37	0.38	0.38	0.39
200	0.47	0.48	0.48	0.49	0.45	0.46	0.46	0.46	0.38	0.39	0.39	0.39

NOTE: This table is for one-step ahead out-of-sample forecasts of an I(1) variable.

Table 3 Mean of the estimated number of factors.

		$\overline{\widehat{p}}$ [for IPC]				$\overline{\tilde{p}}$ [for PANIC]				$\overline{\widehat{r}}$
p	T / N	30	50	100	200	30	50	100	200	30	50	100	200
3	30	2.38	2.41	2.37	2.33	3.00	3.00	3.00	3.00	2.95	3.00	3.00	3.00
	50	2.47	2.47	2.60	2.64	2.99	2.99	2.99	2.99	2.99	3.00	3.00	3.00
	100	2.50	2.65	2.81	2.91	2.98	2.98	2.99	2.99	3.00	3.00	3.00	3.00
	200	2.53	2.73	2.93	2.98	2.98	2.98	2.98	2.98	3.00	3.00	3.00	3.00
2	30	2.05	2.07	1.99	1.93	2.99	2.99	2.99	2.99	2.99	3.01	3.00	3.00
	50	1.89	1.88	1.93	1.93	2.00	2.00	2.00	2.00	3.00	3.00	3.00	3.00
	100	1.87	1.93	1.97	1.99	1.99	1.99	1.99	1.99	3.00	3.00	3.00	3.00
	200	1.87	1.94	1.99	2.00	1.98	1.98	1.98	1.99	3.00	3.00	3.00	3.00
1	30	1.65	1.61	1.47	1.35	2.25	2.24	2.22	2.23	3.01	3.02	3.01	3.00
	50	1.11	1.06	1.08	1.06	0.99	0.99	0.99	0.99	3.00	3.00	3.00	3.00
	100	1.00	1.00	1.00	1.00	0.99	0.99	0.99	0.99	3.00	3.00	3.00	3.00
	200	1.00	1.00	1.00	1.00	0.99	0.99	0.99	0.99	3.00	3.00	3.00	3.00

NOTE: The true number of factors is r = 3, and p is the number of nonstationary factors in the model. The estimator $\widehat{p}$ of p is the estimator based on IPC defined in Proposition 2; its mean over 5000 repeated samples is $\overline{\widehat{p}}$. Similarly, the estimator $\tilde{p}$ of p is based on the PANIC method, and $\overline{\tilde{p}}$ is its mean over 5000 repeated samples; $\overline{\widehat{r}}$ is the mean of the estimate $\widehat{r}$ of the total number of factors r, in Proposition 1.

Table 4 Performance of mixture-FAR relative to AR(4), in terms of $R_{os}^2$.

	$\hspace{0.17em}\log \hspace{0.17em}(\mathit{GDP})$			$\hspace{0.17em}\log \hspace{0.17em}(IP)$
First est. period	Model 1	Model 2	Model 3	Model 1	Model 2	Model 3
1959:Q1 - 1999:Q4	–0.01	0.07	–0.00	–0.14	0.10	–0.21
1959:Q1 - 2005:Q4	–0.08	0.11	–0.01	0.17	0.33	0.11
1959:Q1 - 2008:Q4	0.09	0.16	0.18	0.06	0.28	–0.01

NOTE: Model j (j = 1, 2, 3) is a mixture-FAR model, as defined in (19)–(21). The entry in the table is the $R_{os}^2$ for Model j relative to AR(4). The out-of-sample forecast period extends to 2018:Q4. These results are for expanding windows.

Table 5 Values of $R_{os}^2$ for mixture-FAR compared to the nonstationary-FAR.

First estimation period	log(GDP)	log(IP)
1959:Q1 - 1999:Q4	0.21	0.17
1959:Q1 - 2005:Q4	0.21	0.39
1959:Q1 - 2008:Q4	0.50	0.31

NOTE: The mixture-FAR is Model 1 and the nonstationary-FAR is Model 5. The out-of-sample forecast period extends to 2018:Q4.

Fig. 1 Time series plots of $\hspace{0.17em}\log \hspace{0.17em}(\mathit{GDP})$ and $\hspace{0.17em}\log \hspace{0.17em}(IP)$ for 1959:Q1 – 2018:Q4.

Fig. 2 Left Panel: One-step ahead point-wise 95% prediction intervals for $\hspace{0.17em}\log \hspace{0.17em}(\mathit{GDP})$ using mixture-FAR. The solid line in the middle is the observed values of $\hspace{0.17em}\log \hspace{0.17em}(\mathit{GDP})$. The pair of dashed lines is the asymptotic theory prediction interval; they are easily identified since one curve appears at the top and the other at the bottom, in 2016. The pair of curves with dash and dot, provide the bootstrap prediction interval. Right Panel: The legend is the same as that for the panel on the left, except that GDP is replaced by IP.