all. These are taken as pseudo residuals and used to construct pseudo data for the dependent variable which is:
= n ,, B + C + e^,, (6.21) The same estimation method is then applied to (6.21) to get B* and C*. The
A
above step is repeated 200 times; consequently, the distribution of pseudo errors B*-B,
^ A A
C*-C can be computed and used to approximate the distribution of real errors B-B, C-C (Freedman and Peters (1984b). The formulas for bootstrap estimates of the standard errors of each parameter are calculated in McNown (1991) as:
and
A A
where B and C are the calculated parameter estimates based on the actual data and B*- and C*- are the corresponding estimates using pseudo data of the rephcation. The average standard errors based on conventional asymptotics can be calculated as:
1 200 1 200 and
where SE^ ((3*) and SE^(C*) denote the fomiula standard errors. It is found that average standard errors tend to be biased downwards, as do asymptotic standard errors obtained from estimation of the original data set. Freedman and Peters (1984a) and McNown et al. (1991) argue that analysis of bias in standard errors should be based on comparison of (6.22) and (6.23), since these are based on the same pseudo-data. The resulting differences indicate the magnitude of the biases in the formula standard errors.
ESTIMATION RESULTS
Table 6.1 presents the NL estimation results of cost system (6.10) and summary statistics of the bootstrapping experiment. The NL estimates of the translog cost parameters and their respective standard errors from the asymptotic formulas are reported in the first two columns. An inspection of these two columns reveals that while the parameter estimate
Table 6.1 N L Estimates of Cost Function and Input Share Equations
Initial (NL) Estunates Statistics from Bootstrap
Parameter Parameter (1) Standard Error (2) Mean Parameter (3) Standard Deviation (4) Average Stan- dard Error (5) Cost Function Constant 3.8459*** 0.2690 3.8437 0.0346 0.0346 (Psh/Pwht) -0.1487 0.1629 -0.1432 0.0454 0.0852 (In (PshJPwht)^ -0.2174* 0.1154 -0.2094 0.0313 0.0303 InDj,, -0.0170 0.0319 -0.0164 0.0019 0.0015 0.1099** 0.0425 0.1067 0.0038 0.0021 InDs,, -0.0312 0.0441 -0.0461 0.0212 0.0138 In ( 1 ) 0.6322*** 0.0580 0.6290 0.0175 0.0172 Share Equation Constant -0.1487 0.1629 -0.1432 0.0454 0.0452 In (PshJPwht) -0.2174* 0.1154 -0.2094 0.0313 0.0308 InDj,, 0.0069 0.0132 0.0068 0.0016 0.0016 -0.0644** 0.0184 -0.0633 0.0021 0.0018 InDsht -0.0175 0.0179 0.0252 0.0105 0.0071 -0.6322*** 0.0580 0.6290 0.0175 0.0172
* coefficient significant at 0.10 level; ** coefficient significant at 0.05 level; *** coefficient significant at 0.01 level.
for partial adjustment (k) is highly significant, the coefficient associated with the price variable in the cost share equation is only marginally significant. In order to ensure that the parameter estimates are statistically robust, we need to check test statistics from the bootstrapping experiment based on 200 replications. These are reported in the last three columns of Table 6.1. Column (3) contains the sample mean of the 200 replications of the bootstrap process and can be compared with column (1) as an indication of bias in the NL estimates. The difference between columns (1) and (3) is acceptably small.
The last two columns of Table 6.1 require special attention as they represent the focus of the bootstrapping experiment. Column (4) shows standard errors calculated according to formula (6.22). As expected, these are biased downward in comparison with formula standard errors. Column (5) is average standard deviation, calculated using formula (6.23). Surprisingly, the figures in column (4) are much smaller than those in column (2). This is in sharp contrast to earlier bootstrapping experiments (Freedman and Peters 1984a; Freedman and Peters 1984b; Nainar 1989; McNown et al. 1991), which tended to find that the standard deviation was higher than the formula standard error. However, as mentioned earher, a valid analysis of bias in formula standard errors should be based on a comparison of columns (4) and (5). The difference between the figures in each column is exceptionally small, suggesting that in this case there is little danger in using formula standard errors. W e can therefore have reasonable confidence in the various price elasticities derived from NL parameter estimates. These are set out later in this chapter.
Before tuming to the derivation of price elasticity of demand, a few more comments need to be made on the results reported in Table 6.1. First, the coefficient associated with the price variable in the share equation is still statistically significant at the 0.10 significance level after allowing for bias in conventional standard errors. This indicates that price-induced fibre substitution does exist in the Chinese textile industry.
Second, the estimate of the partial adjustment coefficient ( X ) is 0.63 and is statistically significant, suggesting that the static cost model (6.7) is not an appropriate one for modelling Chinese textile manufacturers' demand behaviour with respect to choice of fibres. The relatively high value of X indicates that the rate of adjustment is quite slow; that is, only about one-third of deviation of observed from optimal shares noted at the beginning of the year is adjusted during the year. Assuming a geometric lag, the mean lag (Griliches 1967, pp. 16-49) can be calculated as:
Mean lag= (1 - ?i) / = 0.58 year V a r i a n c e = ( l - ; ^ ) / ^ ' = 0 . 9 2 year
This tells us that wool textile manufacturers' response has exhibited a geometric lag distribution with a mean of 0.58 years and a variance of about 0.92 years.
Alternatively, as shown by Stewart and Wallis (1989), we can calculate how many time periods are required to accomplish a full adjustment. In general, after n periods, 1-A." of the adjustment will have been accompUshed. If this expression is set to be equal to some value of p, which is a proportion of the desired adjustment, we may solve for n to obtain the number of periods required for adjustment to be complete:
p=\-r
therefore
l o g ( l - p )
n =