In the previous exercise, we examine the TVP model generated with high volatility. However, we will deal with more stable TVP processes in this section. Thus, we will examine performance of Kalman filter on such series. We also consider whether the problems encountered in the previous section repeats in the TVP generated as smooth transitions. We expect Kalman Filter can produce better results since these series are not as volatile as TVP following AR(1) process. We have two different sets of variables for this case. In these sets, the parameter processes differ from each other according to their curvature :
1). Regressor x is generated as driftless unit root as: t 1 x
t t t
x =x− +ε and the TVP process is given by βt =0.5 ( /+ t N)0.5 where N is the sample size. The relation between regressor and dependent variable is governed by the equation:
0.5 y t t t t y = +β x +ε and x, t ε y t
ε are independent standard normal variables. As we mentioned, this type of DGP offers convex functional form for the time varying
0.0015 for the sample size 100. For the sample size 400, these values are given as 1.33 for the mean and 0.009 for the variance of βt.
2). In this case, we consider concave parameter process, which is determined by:
1.5
( / )
t x t N
β =β + where N is the sample size. Moreover, the cointegration relation
is given by y t t t t y =βx +ε and x, t ε y t
ε are independent standard normal variables. In addition, for sample size 100, mean and variance of βt are 1.32 and 0.01 respectively. finally, mean value of βt is 1.11 and variance of βt is 0.04 for sample size 400.
We generated above series with sample size 100 and 400 to check how Kalman Filter performs in estimation of such series. In order to run Kalman Filter recursions with the unknown system matrices, we adopt the state-space form given in equation (31). The parameters of the model are again estimated by ML. Following table exhibits the estimation results and relevant statistics for state vector and RMSE:
Table 16. Identification of the problem: TVP (Smooth Transition) Related Random Walks Case with Unknown System Matrices
Sample Size=100 x~I(1) , β(t)=0.5+(t/N)^γ
mean(at) stdc(at) mean(at1) stdc(at1) coeffvar corr(y/x~at) RMSE
γ=0.5 1.435 0.037 1.428 0.081 0.143 0.407 0.194
γ=1.5 1.317 0.098 1.308 0.147 0.075 0.568 0.195
Sample Size=400
γ=0.5 1.339 0.103 1.336 0.122 0.077 0.454 0.068
γ=1.5 1.116 0.211 1.114 0.218 0.189 0.546 0.059
The above table exhibits the estimation results for sample size 100 and 400. From these tables we can see that Kalman Filter does not generate state estimates and predictions as volatile as in the TVP generated as the AR(1) processes. This picture possibly appears because of the absence of the noise term in the parameter process. We can also see that the parameter γ slightly affects the volatility in a and the t mean value of a . On the other hand, the mean and variance of t a and t at t| 1− are computed very close to the mean and the variance of true parameter process βt. Hence, Kalman filter creates the estimated and predicted state vector around the mean of βt with almost same variance. Additionally, the histograms of all of the data
generation processes (Tables 30-33 in appendix B) exhibits similar picture for ML estimates of α, T , µ and σq. However, the shapes of distributions are clearer for sample size 400. From these histograms, it can be observed that the estimated value of T is accumulated around 1. We also see some outliers for T around zero, whereas these outliers do not change the analysis too much. However, there is an interesting result for µ. The ML estimator µMLE is approximately zero for all sample size and γ . Thus, we can conclude Kalman Filter ignores the constant term in the data generation of βt and include the time invariant part of the state in the time variant part. Furthermore, the table 16 reports smaller RMSE than table 14 and 15 for all cases. Thus, Kalman filter produces better prediction for y in the estimation of the t time varying parameters with low volatility. This is an expected result since the volatility of the true parameter vector also increase the variation in the dependent variable. As a consequence, increase in volatility of y reduces traceability of the t underlying model by Kalman Filter, since noisy data will cause Kalman Filter underperforms.
On the other hand, the figures 33 (for γ=0.5) and 35 (γ=1.5) illustrate the results of the estimation of the time varying parameters in the smooth transition model with Kalman Filter where the sample size is equal to 100. These figures verify above comments. Therefore, Kalman filter algorithm with ML estimation yields very good prediction of the data. Additionally, in all cases, Kalman Filter algorithm estimates the time varying parameter very close to the actual parameter process. Figures 34 and 26 exhibits the relation between the estimated state vector and the actual parameter process for the different choice of γ. The predicted and estimated state vectors seem to follow exactly the same path as βt. Therefore, performance of Kalman filter is excellent in the estimation of the time varying parameter models with smooth transitions. The only problematic situation is that the ML estimate of µ is miscomputed, whereas this does not impact the estimation performance.
In conclusion, this chapter includes the simulations with 4 different types of the data generation process. In all of these simulations, Kalman Filter performs well in predicting the dependent variable y with the knowledge of the regressor x .
namely the unrelated random walk series. In the unrelated random walk case, we expect that Kalman Filter detect the inexistence of relation between variables. However, this is not the case. Instead, Kalman Filter produces a nonsense state vector that links y and t x . Moreover, the emergence of meaningless state vector is t accompanied with highly volatile predicted and estimated state sequences. The high variations in a and t at t| 1− do not help us detect spurious regression yet, since the time varying parameter models can also yield high variation in the state according to variance of the TV parameter. In addition, Kalman Filter underperforms in the estimation of TVP following AR(1) processes. We encounter with the inaccuracies in estimation of βt. We think these imprecision potentially emerge because of high volatility of the true parameter process, as we do not face with the same problem in time fixed or smooth transition models. These models have excellent performance in capturing the true model. On the other hand, we encounter very different estimates for T in each data generation process. While ML estimate of T is computed as 0 in the linearly related random walk case, this parameter is estimated as approximately 1 in the estimation of independent random walks and TVP model with smooth transitions. Therefore, the estimated value of T does not give clue about the existence of the spurious regression.