Public capital and private production in Australia.

• Author: Otto, Glenn D.

1. Introduction

   Recent empirical research by Ratner [31], Aschauer [4; 5] and others seeks to demonstrate that publicly provided capital should enter as a complementary input to private production.(1) The general approach of these studies is to specify and estimate a functional form for private production which is dependent upon both private inputs to production and some measure of public capital. With few exceptions, see for instance Holtz-Eakin [18], across a variety of economies and at different levels of aggregation, public capital is shown to be a significant input to private production.

   This paper uses a newly constructed quarterly data set for the Australian economy to examine two specific criticisms levelled at many of the previous empirical studies. The first is the possibility that the estimated relations reflect a spurious correlation between variables with purely coincidental low frequency movements; that is, the variables are non-stationary and the regressions are spurious. This criticism is addressed by Aschauer [3] where he argues that his previous results are valid even if the variables are non-stationary. More recently, Clarida [10] and Lynde and Richmond [21] use techniques suitable for non-stationary variables to confirm Aschauer's [4] results for the United States as well as for a number of European countries. We also employ co-integration techniques, those developed by Phillips and Hansen [29] and Hansen [17], to estimate a production function for the private economy in Australia during the post-war period.

   The second issue we address is the question of causality between public capital and private production. Aschauer [4] argues that causality runs from public capital to private output (or private factor productivity). However, an equally plausible interpretation of the estimated relationships is that they reflect the response of public investment to private production; in other words, public capital stocks are possibly endogenous. This endogeniety may arise from various political constraints on public expenditure or, alternatively, the relationship may reflect aspects of the public decision-making process where public investment decisions are a response to private output growth or possibly private investment. To examine this second criticism, we employ vector autoregression (VAR) techniques associated with Sims [33] to examine the effect of public capital on private sector variables and also to examine whether there is any feedback from the private sector variables to the public capital stock. Unlike previous studies, which use annual data, our quarterly data provides us with a sample size large enough to effectively use these techniques.

   The paper proceeds as follows. Section II presents the basic representation of private production used in the empirical analysis. In section III, single equation cointegration techniques are used to estimate the long run relationship between private production and private and public inputs. A VAR analysis is used in section IV to examine the dynamic interactions among the variables of interest. Section V concludes.

2. Representations for Private Production

   Private production is represented using a Cobb-Douglas production function. We allow for three inputs: private capital, public capital, and private labor. The justification for treating public capital as an input to private production is discussed by Arrow and Kurz [2]; the Cobb-Douglas functional form is employed for its suitability in empirical analysis. Four alternative specifications for private production are considered:

   y - k = [[Alpha].sub.n1](n - k) + [a.sub.1] (1)

   y - k - g = [[Alpha].sub.n2](n - k - g) + [a.sub.2] (2)

   y - k = [[Alpha].sub.n3](n - k) + [[Alpha].sub.g3] (g - k) + [a.sub.3] (3)

   y - k = [[Alpha].sub.n4](n - k) + [[Alpha].sub.g4] [multiplied by] g + [a.sub.4] (4)

   where y is private sector output, n is private sector labor, k is private sector capital, g is public capital and [a.sub.i], i = 1 . . . 4, is a non-observable measure of technology. All variables are in logarithms.

   These four models represent alternative ways of treating public capital as an input to private production. Model (1) assumes constant returns to scale (CRS) between private inputs and no role for public capital. Model (2) includes public capital as an input to production but as a perfect substitute for private capital; the model assumes CRS across all inputs. In many studies of aggregate production, for example the real business cycle studies initiated by Kydland and Prescott [20], it is common to aggregate public and private capital stocks, effectively treating them as perfect substitutes. As far as private production is concerned, comparison of model (2) with the other specifications provides some information about the suitability of doing so. Models (3) and (4) treat public capital as a complementary input. Model (3) assumes CRS across all inputs and, necessarily, decreasing returns to scale in private inputs. This representation is suitable if public capital is not a pure public good. Model (4) assumes CRS between private labor and capital but allows increasing returns to scale (IRS) across all three inputs. This representation is suitable if public capital is a non-rival input in private production.

   The latter two models form the basis of much of the empirical literature on public capital provision. Ratner [31] estimates model (3) for the United States and obtains an estimate for the output elasticity of public capital of 0.06. Aschauer [4] estimates both (3) and (4) for the United States, finding greatest support for the CRS specification. His elasticity estimates are in the order of 0.40, notably larger than Ratner's estimates. Otto and Voss [25] estimate models (3) and (4) for the Australian economy using annual data and obtain results very similar to Aschauer's.

   A feature of Aschauer's results, and by implication our results for Australia, is that the elasticity estimates imply very high rates of return to additional investment in public capital compared to estimates obtained from project based cost-benefit studies. Consequently, a number of authors, for example Winston and Bosworth [34], have argued that the public capital marginal productivity estimates from aggregate production functions are implausibly large.(2)

   Thus we are interested in seeing if these large elasticities are robust to the use of econometric techniques which can account for non-stationary series.

3. The Long-Run Elasticities

   In this section, we use techniques suitable for non-stationary data, specifically integrated processes, to estimate each of the four models of private production. We first establish that the variables of interest are indeed integrated of order one, 1(1), and then seek to determine if the input variables and private production are cointegrated. If these variables are cointegrated then we can reject the argument that previously estimated relationships in levels are spurious and in addition obtain consistent output elasticity estimates from the low-frequency components of the data.

   Each model is estimated using quarterly time series data for Australia over the period 1959:3 to 1992:2. The data we use is not directly available from published sources. Measures of output, capital stock and hours worked for the private sector are constructed along with a series for government capital, which includes both general government capital and the capital stock of public trading enterprises. Full details on primary data sources and the method of construction are provided in an appendix.

   Since the technology variable [a.sub.i] is itself not observable, we follow standard practice and treat this as part of the disturbance term. This creates the additional possibility that evidence of no cointegration between the other variables may be indicative of an integrated technology process and not a rejection of the model itself.(3) Pursuit of these matters, however, is beyond the scope of this paper and we are limited to simply recognising the possibility.

   The first step is to perform tests for unit roots on all observable variables in equations (1) to (4), both in levels and in first-differences, to determine if these variables are: I(1). Table I presents augmented Dickey-Fuller (ADF) tests. The evidence suggests that the variables y - k, y - k - g, n - k, and n - k - g are I(1) variables. The results of the ADF test for the variables g - k and g are not as sharp. Whether we can view these two variables as being stationary in first-differences is sensitive to the choice of lag length in the ADF test regression. Of the two series, the latter seems least likely to be first-difference stationary. In fact, a two-sided ADF test leads to the rejection of the null hypothesis that g is I(1) in favor of the alternative that it follows an explosive autoregressive process. This indicates that g behaves more like an I(2) than an I(1) process and leads us to suspect that model (4), the IRS specification, is unlikely to perform well on the quarterly data set.(4)

   Given these results which suggest that most of the variables in models (1) to (4) are I(1), we now estimate each of the models and test whether any of them represent valid cointegrating relationships. To estimate these models we embed them in a triangular representation for cointegrated systems of the following form:

   [Z.sub.t] = [mu] + [Beta]t + [X[prime].sub.t] [Alpha] + [u.sub.1t] (5)

   [Delta][X.sub.t] = [Delta] + [u.sub.2t] (6)

   where [Z.sub.t] is the dependent variable in models (1) to (4) and [X.sub.t] is a vector of the stochastic regressors. A set of conditions on the vector of error terms [u.sub.t] = ([u.sub.1t], [u.sub.2t])[prime] for (5) and (6) to be a valid cointegrating system are given by Park and Phillips [28]; these are assumed to be satisfied here.

   Table I. Augmented Dickey-Fuller Tests

   Variable Lags (m): 1 2 3 4

   y - k -2.83 -3.13 -3.14 -3.06 n - k -0.86...