Publications Empirical Studies HOCKE, KYN, WAGENER  PRODUCTION FUNCTIONS ESTIMATES FOR SOVIET INDUSTRY

3.      Some Methodological Problems

3.1 The already mentioned multicolinearity makes it difficult if not totally impossible to separate the influence of technical progress from the influences of economies or diseconomies of scale and of capital and labor elasticities of output.

Let us illustrate this well known problem on the following simplified example. Suppose that the production function has a form (1b) and that both capital and labor grow at constant rates k and l respectively. The production func­tion can be then written in the form:

 (9) Y = A0 e r t K0beb k t L0ge g l t + e

where ee is the random disturbance.

After simple manipulations (9) can be rearranged into

 (10a) lnY = a + gat + e

where

 (l0b) a = lnA0 + b lnK0 + g lnL0

and

 (10c) ga = r + bk + gl

The only obtainable information about the rate of technical progress r and the elasticities of output b and g is that their linear combination with weights l,  k and l  must be equal to the average actual rate of growth g. This means that only the mutual relation of r and g given by (10c) is known but the correct values of individual parameters re­main undetermined. In other words, any set of parameters (r, b, g) which satisfies (10c) would give an equally good fit.

Suppose for example that we choose purely arbitrarily the parameters b* and g*. The estimated rate of technical change would then be

 (11a) r* = ga - l - b*k - g*l

For constant returns to scale (11a) can be simplified into

 (11b) r* = ga - l - b*(k - l)

The relations (11a) and (11b) represent the well known “trade off” between the estimated rate of technical chance and the estimated parameters b and g. It is apparent that higher estimates of b and g imply lower estimates of r.

Each set of arbitrary parameters r*, b* and g*   satisfying (10c) is likely to produce  different estimate of the equilibrium rate of growth. Substituting  (11a) into (5a) we get

 (12) ge = (ga - l - b*k )/(1 - b*)

Formula (12) shows that if multicolinearity is very high the growth potential of the economy cannot be uniquely determined. It can be expressed only conditionally with regard to the assumed value of b and implicitly of r.

3.2. It may be interesting to asses the bias in the estimated rate of technological change r and in the estimated growth potential ge resulting from the wrongly estimated parameters b and g. The bias depends on the relation of the actual growth path, from which the observed data are derived, to the hypothetical equilibrium growth path.

For the sake of simplicity, let us assume the nonstochastic case and suppose that we know the explicit form of the true production function

 Y = A0e .03t K .4 L .6

Let us further assume that the labor force grows at a con­stant rate l = .02. The equilibrium rate in this case is equal

 ge = l + r/(1 - b) = .02 + .03/(1 - .4) = 0.07

Let us now investigate three cases.

Case I:  The actual rate of growth of output and the rate of growth of capital are equal, i.e. the economy was on the equilibrium path

 ga = k = 0.07

Case II: The rate of growth of capital is higher than the actual rate of growth of output  which can be derived from the true production function

 k = 0.09 ga = 0.078

Case III: The rate of growth of capital is smaller than the actual growth

 k = 0.05 ga = 0.062

Using (11b) we can calculate the estimated rates of technical progress r for various possible values of the parameter b*.

 r*I = .05 -.05  b* r*II = .058 - .07 b* r*III = .042 - .03 b*

Diagram 1 shows the different interelationships of  r* and b*    which naturally intersect at the true values of r and b

.

Using either (5b) or (12) we now can find the equilibrium rate of growth ge = ge (b*) corresponding to each value of b* and implicitly also of r* Table 3 and Diagram 2 show the results.

Table 3

Case I               CaseII             CaseIII

b*              r*I             ge*I             r*II     ge*II   r*III         ge*III

0.0             .05             .07               .058           .078          .042           .062

.2               .04             .07               .044           .075          .036           .065

.4               .03             .07               .03             .07            .030           .07

.6               .02             .07               .0l6            .06            .024           .08

.8               .0l              .07               .002           .03            .018           .11

.9               .005           .07             -.005          -.03            .0l5            .17

.95             .0025         .07             -.0085        -.15            .0135         .29

We see that   1. in case I the growth potential is unbiased even with high estimation errors in b*.

2. if the economy was above the equilibrium path in the ob­servation period (case II) an estimated capital elasticity of output higher than its true value will lead to an under­estimation of the growth potential: b* > b  =>    ge*  <  ge  and if the economy was below the equilibrium path during the observation period (case III) an underestimation of the capital elasticity will result in an overestimation of the growth potential: : b* < b  =>    ge*  >  g

3. the degree of bias in g*e resulting from the incorrectly estimated b* is likely to be relatively small if b* < b  but it can become extremely large if  b* > b.

3.3. The problem of multicolinearity may be avoided if the macroeconomic production function is estimated directly from sectoral data rather than from the aggregated time series Intuitively, this is a reasonable approach. Why should we lose the precious degree of freedom by aggregating sectoral data before we start with regressions. No new in­formation can be gained by aggregation, the only result being a reduced number of observations. The suggested method is usually described as a combination of cross-section and time-series analyses because it estimates the production functions from the “time-series of cross-sectional data”.

The regression equations have an ordinary form, except that two subscripts, one for sector and one for the year are being used for each observation. For example the regression equation for the standard Cobb-Douglas production function with the constant rate of technical change would be

 (13) lnYit = a + b1n Kit + g ln Lit  +  rt + eit

The straight application of the ordinary least squares method to the pooled cross-section and time-series data may give statistically significant coefficients, however, the R2  is usually very low. This can be improved by introducing sectoral dummy-variables, defined in the following way:

 (14) Dijt  =  1 for i = j Dijt = 0 for i ¹  j

The regression equation has then the form

 (15) lnYi t = Sj aj Dijt + b1n Kit + g ln Lit  +  rt + eit

The sectoral dummy variables capture the sectoral differences in the level of the factor productivity, which means that each sector has its own intercept  ai   but they all do commonly share the slopes b and g and the shift coefficient r.

3.4. Sometimes it is impossible to maintain simultaneously the assumptions that the rate of the technological change is constant and the elasticity of substitution is equal to one. Suppose, that we observe a declining trend in the rates of growth of output, but relatively constant rates of growth of capital and labor. From the equation

 (16) dY/dt (1/Y) = r  +  hK  dK/dt (1/K)  + hL  dL/dt (1/L)

it is clear that the observed trends may be explained either by declining r(t) or by changes in the elasticities  hK  and hL . The first approach would lead to the estimation of Cobb-Douglas production function with the trend in technological change, for example aproximated by (6a). The other approach leads to the estimation of production functions with a nonunitary elasticity of substitution, for example the CES production function (1a). Both approaches are likely to improve the fit, however, it may be very difficult to establish,  which of the approaches is better. The observed facts can usually be explained equally well by a declining rate of technological change or by a low elasticity of substitution.

3.5 Another problem appearing in the time series approach as well as in the estimations from the pooled cross-section and time-series data, is the serial correlation in errors. It is known that with serially correlated errors the ordinary least squares method (OLS) does not yield the BLU estimator. However, the properties of the estimates can be improved if the variables are transformed by the first order autore­gressive coefficients. Because the OLS estimates of auto­regressive coefficients are biased, it is sometimes necessary to repeat the transformations several times before the serial correlation is fully eliminated.

A similar procedure can be applied also in estimations from pooled cross-section and time-series data. In this case it is useful to assume a second order autoregressive process and to estimate the coefficients separately for each cross-sectional unit (industrial branch).

3.6 The parameters of the CES production function were estimated by a nonlinear regression program based on Marquardt’s algorithm (see Marquardt, D.W.(63)). (Technical description of the Marquardt’s algorithm was omitted )

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