State-dependent biases and the quality of China’s preliminary GDP announcements

Abstract

This paper investigates whether and how the systematic forecast errors of the quarterly GDP announcements in China depend on the state of the economy. Our contribution is both theoretical and empirical. On the theoretical side, we extend the predictive threshold regression of Gonzalo and Pitarakis (J Bus Econ Stat 35:202–217, 2017) by incorporating a time-varying and state-dependent threshold, which is a function of macroeconomic variables that affect the separation of regimes. On the empirical side, we apply our model to assess the quality of China’s preliminary GDP data. Our empirical results show that there exist forecast biases in the preliminary GDP data conditional on the state of the economy. Our results also lean toward supporting that there exist behavioral biases of underestimation and over-reaction to new information during periods of relatively good state. These results suggest some scope to improve the accuracy of the preliminary GDP data based purely on econometric models.

Appendix: Monte Carlo Simulations

In this Appendix, we conduct Monte Carlo simulations to investigate the finite sample performances of the proposed estimation and testing procedures. We provide simulation evidence to support that overlooking time-varying features in the predictive threshold model can result in biased estimates and a significant power loss of the test for threshold effect. We also present a number of simulation experiments calibrated on the empirical example to demonstrate the accuracy of the parameter estimates and the size and power properties of the tests in the empirical example. First, we consider the data generating process (DGP) specified as

$$\begin{aligned} {y_{t +1}} = \left\{ \begin{array}{l} {\alpha _1} + {\beta _1}x_t + {u_{t+1}},\quad \text{ if }\;\; {q_t} \le {\gamma _t}\\ {\alpha _2} + {\beta _2}x_t + {u_{t +1}},\quad \text{ if }\;\; {q_t} > {\gamma _t} \end{array} \right. , \end{aligned}$$

(A1)

where $\gamma _t = {\gamma _0} + {\gamma _{1}}z_{t}$. We assume a normally distributed error $u_t\sim i.i.d. N(0, 1)$, $q_{t}$ and $z_t\sim i.i.d. N(0,1)$, $x_t\sim i.i.d.N(0,4)$. The sample size T is set as 50, 100, 200 or 500.

Table 5 Estimates of the threshold model with a time-varying threshold

In the simulations for the estimation procedure, we set $\alpha _1=\beta _1=1$, $\alpha _2=\beta _2=2$, $\gamma _0=0$ and $\gamma _1=1$. The number of replications is 2000. The simulation results are reported in Table 5, in which we compare the proposed estimation procedure with the estimation procedure assuming a constant threshold. Panel A of Table 5 presents the summary statistics (i.e., mean and standard deviation) for the estimates based on the estimation procedure proposed in Sect. 2. The mean of each parameter comes close to its true value, and the standard deviation becomes small as the sample size increases, indicating that the accuracy of the estimates improves. Overall, for finite sample sizes, in terms of bias and standard deviation the performance of the proposed estimation procedure is generally satisfactory. Panel B of Table 5 reports the summary statistics (i.e., mean and standard deviation) for the estimates based on the estimation procedure assuming a constant threshold. These simulation results show that, if we ignore time-varying features in the threshold, the estimates of the parameters are seriously biased and with large standard errors. Furthermore, these biases could not disappear as the sample size increases, implying that ignoring time-varying features in the threshold would lead to misleading empirical results.

In assessing the finite sample performance of the proposed test statistics, the rejection frequencies under the DGP with $\{\alpha _1=\alpha _2=\beta _1=\beta _2=1\}$ and $\{\alpha _1=\beta _1=1, \alpha _2=\beta _2=2\}$ are the sizes and powers of the test statistic $F_{1C}$ and $F_{1T}$ defined in Sect. 2, respectively. The rejection frequencies under the DGP with $\gamma _1= 0$ and $\gamma _1=1$ are the size and power of the test statistic $F_{2T}$ defined in Sect. 2, respectively. Table 6 reports the simulation results. In examining the size and power of $F_{1C}$, we consider two cases: Case 1 with a constant threshold $(\gamma _0, \gamma _1)=(0,0)$ and Case 2 with a time-varying threshold$(\gamma _0, \gamma _1)=(0,1)$, which are reported as $F_{1C}$ and $F^*_{1C}$, respectively. The simulation results show that the power of the test statistic $F_{1C}$ for threshold effect would decrease with the inclusion of a time-varying threshold. Turning now to the test statistics $F_{1T}$ and $F_{2T}$, the simulation results support that the size and power properties of the test statistics are generally satisfactory even when the sample size is small (e.g., $T=50$). To summarize, we conclude that, if we consider the time-varying threshold correctly, the test statistics work well in finite samples; however, ignoring time-varying features in the threshold would lead to a significant power loss of the test $F_{1C}$.

Table 6 Size and power of the test statistics $F_{1C}$,$F_{1T}$and $F_{2T}$

Given the small sample size in the empirical part, it is useful to investigate the accuracy of the parameter estimates and the size and power properties of the tests in the empirical example. Moreover, the proposed bootstrap approach does not account for the time series nature of the observations in empirical applications, and highly persistent regressors will distort the test statistics, as discussed in Hansen (2017). Unfortunately, there is little guiding theory for the underlying econometrics in these instances. We therefore present a number of simulation experiments calibrated on the empirical investigation.¹³ Our data generating processes are

$$\begin{aligned}&{A_{t,h+1}} = \left\{ \begin{array}{l} {\alpha _1} + {\beta _1}F_t + {u_{t,h+1}},\quad \text{ if }\;\; {q_t} \le {\gamma _t}\\ {\alpha _2} + {\beta _2}F_t + {u_{t,h+1}},\quad \text{ if }\;\; {q_t} > {\gamma _t} \end{array} \right. , \end{aligned}$$

(A2)

$$\begin{aligned}&A_{t,h+1} -F_t= \left\{ \begin{array}{l} {\alpha _1} + {\beta _1}(S_{t,h}-F_t) + {u_{t,h+1}},\quad \text{ if }\;\; {q_t} \le {\gamma _t}\\ {\alpha _2} + {\beta _2}(S_{t,h}-F_t) + {u_{t,h+1}},\quad \text{ if }\;\; {q_t} > {\gamma _t} \end{array} \right. , \end{aligned}$$

(A3)

in which $\gamma _t=\gamma _0+\gamma _1{ inflation}_t+ \gamma _2 \overline{{ growth}}_{t-4}$, and we set the parameters as reported in Tables 3 and 4 to match the estimates in the empirical investigation. For example, in the case of model (A2) we set $\alpha _1=7.683, \beta _1=0.822, \alpha _2=2.800, \beta _2=0.999$, $u_t\sim N(0,18.783)$ and $\gamma _t=0.162-0.3{ inflation}_t+ 1.9 \overline{{ growth}}_{t-4}$ to match the estimates for the headline GDP in Table 3. We fix $F_t, { inflation}_t$ and $\overline{{ growth}}_{t-4}$ to equal the empirical values used in Sect. 3, and set $T=60$ as in the empirical example. Setting $F_t$ to equal the sample values can force the simulations to assess the performance in a setting with the serial correlation properties of the observed GDP data. We generate 2000 samples from this process to evaluate the properties of the proposed estimation procedure. The assessment of the accuracy of the parameter estimates is reported in Table 7, in which we compute the summary statistics (i.e., mean and standard deviation) for the estimates. The mean of each parameter comes close to the estimates reported in Tables 3 and 4, and the magnitude of standard deviation is relatively small, implying that the estimation procedure works well.

Table 7 Accuracy of the parameter estimates based on the data generated from (A2) and (A3)

To evaluate the size and power properties of the proposed test statistics $F_{1C}, F_{1T}$ and $F_{2T}$, we use the same parameterization as above. Take $F_{1T}$, for example, to evaluate size, we set the parameters of model (A2) as $\alpha _1=\alpha _2=3.076, \beta _1=\beta _2=0.980$ (null hypothesis), and $u_t\sim N(0, 44.972)$ for the headline GDP case to match the estimates based on the linear model ${A_{t }} =\alpha + \beta F_t + {u_{t}}$.¹⁴ Meanwhile, to evaluate the power of the test, we set $\alpha _1=7.683, \beta _1=0.822, \alpha _2=2.800, \beta _2=0.999$ (alternative hypothesis), $u_t\sim N(0,18.783)$ and $\gamma _t=0.162-0.3{ inflation}_t+ 1.9 \overline{{ growth}}_{t-4}$ to match the estimates for the headline GDP in Table 3. We can evaluate the size and power properties of the test statistics $F_{1C}$ and $F_{2T}$ using similar techniques. The simulation results with the block length $b=4$ are reported in Table 8. At the 5% nominal significance level, the simulation results show that the tests have reasonable size for all cases, implying that spurious rejections could not occur in the empirical example. Turning now to the power properties, the rejection frequencies are low in the cases where we could not reject the null hypothesis of no threshold effect in Table 2, as the magnitude of threshold effect (the difference between good and bad states) is relatively small in these cases; however, the rejection frequencies are reasonably high in the cases where we can reject the null hypothesis in Table 2. That is, the power of the test statistics would increase as the magnitude of threshold effect becomes large, which is consistent with the result in Table 6. These simulations indicate that the testing results have reasonable accuracy in the empirical investigation.

Table 8 Size and power properties of the tests based on the data generated from (A2) and (A3)