Sample Size and the Optimal Approximation Order

Here we explore the relation between the sample size and the optimal approximation order.

The optimal approximation order is defined here as the expansion order of the consumption Euler equation that yields the minimum MSE of the β₂ estimates. Intuitively, we would expect that β₂ can be estimated more efficiently and the optimal approximation order would be greater as we increase the sample size. The reason is that the increased sample size provides more information in estimating the structural parameters, and that the optimal approximation order is now more ‘convergent’ to the true order of infinity.

We now present the Monte Carlo results of the different sample sizes and approximation orders in Table 5.⁹ In this benchmark case, we set ρ = 3. And the sample size N is set to be 1000, 2000, 5000, 8000, 10000, 20000, 50000, 80000, and 100000, respectively. Estimation results are reported up to the 10th order.¹⁰

We first look at the bias of the β₂ estimates. As is also evident from Figure 1, the approxi-mation bias can be significantly reduced if the proper approxiapproxi-mation order is used, except for the case where N=1000. The approximation bias seems to be like a ‘inverse U’ shaped curve in the figure. This means that the bias declines first as the higher-order consumption moments are added into the regression. Nevertheless, when the bias reaches its minimum which is fairly close to zero, the bias then increases dramatically as even more higher terms are included.

The former results from that the error term are now more ‘purged’ and are therefore much more orthogonal to the regressors when higher-order terms are introduced. The later then results from that as more and more higher-order moments are included into the model, few further information on the consumption behavior (distribution) can be provided. The resulting multi-collinearity thus leads very inaccurate β₂ estimates.

We also find that the increase in sample size does not help in reducing the approximation bias of the 2nd-order approximated estimations. They consistently yield estimates that are close to 1.300, which implies a bias about -0.7. The advantage of increasing the sample size is quite obvious, however, when the higher-order moments are introduced into the model. The minimum of the bias (in absolute value) can be reduced from 0.299 when N=1000 (7th-order approximation) to 0.025 when N=10000 (8th-order approximation) and to 0.018 (7th-order approximation) when N=100000.

The standard errors of the β₂ estimates are depicted in Figure2. Apart from what we have mentioned before that there is a great reduction in estimation efficiency when higher-order

9The OLS estimation results are not reported because adding higher-order terms is of no use in reducing approximation bias. Since we are interested in the potential benefit of adding higher-order terms, we thus choose not to report the OLS results. Moreover, as the IV2 estimation results are qualitatively similar to the IV1 results, to save space, we do not report them either.

10With the 16 instrumental variables we are using, we can actually approximate the Euler equation up to the 16th order. But the multi-collinearity problem becomes so severe that the variance ofβ2estimates become very huge. We thus choose not to report them in the table.

2 3 4 5 6 7 8 9 10 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

|bias|

order N=1000

N=100000

N=10000

Figure 1: Biases of Different Sample Sizes and Approximation Orders

2 3 4 5 6 7 8 9 10

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Std.

order N=1000

N=100000

N=10000

Figure 2: Standard Deviation of the β₂ Estimates from Different Sample Sizes and Approxi-mation Orders

Table 5: β₂ Estimates with Different Sample Sizes and Approximation Orders (ρ = 3)

Order N=1000 N=2000 N=5000 N=8000 N=10000 N=20000 N=50000 N=80000 N=100000

2 1.294 1.296 1.296 1.302 1.303 1.308 1.308 1.309 1.310

(0.158) (0.148) (0.100) (0.071) (0.063) (0.045) (0.032) (0.021) (0.019)

3 1.605 1.584 1.562 1.560 1.552 1.538 1.502 1.483 1.476

(0.192) (0.138) (0.105) (0.095) (0.100) (0.089) (0.084) (0.084) (0.077)

4 1.511 1.662 1.755 1.768 1.768 1.738 1.680 1.674 1.653

(0.460) (0.331) (0.235) (0.210) (0.221) (0.207) (0.161) (0.152) (0.130)

5 1.236 1.495 1.767 1.845 1.865 1.934 1.934 1.913 1.904

(0.491) (0.424) (0.327) (0.293) (0.281) (0.259) (0.293) (0.326) (0.321)

6 1.400 1.476 1.677 1.766 1.892 1.899 2.010^∗ 2.046 2.046

(0.506) (0.396) (0.326) (0.298) (0.303) (0.245) (0.285) (0.327) (0.336)

7 1.701^∗ 1.814^∗ 1.820 1.854 1.852 1.971 1.922 1.978 1.982^∗

(0.627) (0.437) (0.336) (0.292) (0.281) (0.217) (0.257) (0.283) (0.281)

8 1.164 1.696 2.019^∗ 2.036^∗ 2.025^∗ 1.985^∗ 1.960 1.975 1.912

(0.969) (0.883) (0.549) (0.394) (0.438) (0.274) (0.184) (0.207) (0.217)

9 1.199 1.139 1.587 1.892 2.126 2.037 1.970 1.980^∗ 1.980

(5.780) (0.903) (1.907) (0.804) (1.114) (0.848) (0.348) (0.637) (0.200)

10 1.087 1.316 1.329 1.596 1.564 2.150 1.923 1.963 1.878

(4.099) (1.610) (5.019) (3.845) (1.390) (2.544) (0.889) (1.430) (1.597) Notes. Figures reported in the table are the average estimates of β2 among 1,000 replications. Standard

errors in parentheses. β₂ should be equal to 2 whenρ = 3.

∗ The approximation order that yields the minimum absolute value of bias for each sample size.

consumption moments are introduced into the model, the relation between the estimation efficiency and the sample size is also obvious. For most approximation orders, the estimation efficiency can be enhanced when the sample size is enlarged. The reason is that the greater sample size provides more information of the parameter of interest, and the parameter can thus be estimated more precisely.

We now turn to the MSE criterion that accommodates both the concern over the unbiased-ness and the estimation efficiency. The MSEs of the β₂ estimates of different approximation order and sample sizes are summarized in Table 6. For each sample size, the ‘optimal approx-imation orders’ that yields the minimum MSE are labeled with ‘∗’ in the table. The result is that when we increase the sample size, the optimal approximation order increases. A smaller MSE can be achieved as well. The minimum of the MSE can be reduced from 0.193 when N=1000 to 0.097 when N=10000, and to 0.040 when N=100000.

This pattern can be seen clearly in Figure 3. The MSEs seem to exhibit a U-shaped

Table 6: MSE of β₂ Estimates (ρ = 3)

Order N=1000 N=2000 N=5000 N=8000 N=10000 N=20000 N=50000 N=80000 N=100000

2 0.524 0.517 0.505 0.492 0.490 0.481 0.482 0.477 0.477

3 0.193^∗ 0.192^∗ 0.203 0.203 0.211 0.222 0.255 0.274 0.280

4 0.452 0.224 0.116^∗ 0.097^∗ 0.103 0.119 0.129 0.130 0.138

5 0.824 0.436 0.161 0.110 0.097^∗ 0.071 0.091 0.114 0.113

6 0.616 0.432 0.210 0.143 0.103 0.070 0.081 0.109 0.115

7 0.483 0.225 0.146 0.106 0.109 0.054^∗ 0.072 0.081 0.079

8 1.638 0.872 0.301 0.157 0.193 0.075 0.038^∗ 0.044^∗ 0.051

9 34.053 1.557 3.808 0.659 1.255 0.720 0.122 0.406 0.040^∗

10 17.634 3.060 25.638 14.950 2.121 6.495 0.796 2.047 2.566

Notes. Replication for 1,000 times.

∗ The ‘optimal approximation order’ with the minimum MSE.

pattern when we increase the approximation order. This means that there does exist some approximation order that yields the minimum MSE. The idea that we do not suggest unlimited expansion of the consumption Euler equation can thus be verified. We also find that with a greater sample size, the U-shaped MSE curve shifts rightwards and downwards, implying a smaller minimum MSE achieved with a greater approximation order.

To conclude the subsection, we depict the optimal approximation order and the corre-sponding MSE of the different sample sizes in Figure 4. As is evident from the figure, the optimal approximation order increases when a greater sample size is used. The sharp decline in MSE reveals the fact that a smaller bias can be achieved and the efficiency gain from increased sample size.