5  Modelling

In this section we develop and describe a simple stylised statistical error components model that is (broadly) consistent with some of the stylised facts pertaining to the autocorrelation properties of the data described in Section 4, particularly Section 4.1. There is a large literature on modelling firm productivity dynamics (e.g. see Sutton, 1997, for an overview). However, as with the previous results, we do not explicitly consider firm entry and exit decisions and effects. Rather, our modest objective here is simply to calibrate this model to these stylised facts, in order to provide a possible interpretation of the sources of variation observed in the GST-based measures of labour productivity. We also remain largely agnostic on the extent to which the observed variations reflect (true) productivity variation or data quality issues.

The ‘stylised facts’ of the data that the model will attempt to replicate can be summarised as follows:

1. Autocorrelations in sales per hour (S), purchases per hour (P), and labour productivity per hour (^lP) each range from (trivially) 1 contemporaneously, then fall sharply to around 0.6 on average at 1-year lags, and then gradually to around 0.2 at 9-year lags.
2. Cross-autocorrelations between S and P are also high (about 0.95 on average) contemporaneously, fall sharply to about 0.6 at lag-1, and then gradually to around 0.2 at lag-9.
3. Cross-autocorrelations between S and ^lP are about 0.4 contemporaneously, and then fall gradually to about 0.1 at lag-9.
4. Cross-autocorrelations between P and ^lP are about 0.2 contemporaneously and fall to near-0 by lag-9.

5.1  The model

The model we develop here focuses on firm sales per hour (S) and purchases per hour (P), and treats labour productivity (^lP) as simply the difference between S and P. There are two basic characteristics of the model. First, in a univariate context, our statistical modelling of S and P allows each to have three components of firm-level variation: a time-invariant permanent component, a persistent (but declining) component, and a purely transitory component. Loosely speaking, the first two components may be thought of as being associated, respectively, with permanent differences in business activity across firms, perhaps due to firm-specific technologies, patents, and location and/or human capital (dis)advantages; persistent, though non-permanent business activity shocks, perhaps due to cyclical or other temporary shocks (e.g. droughts and/or exchange rate movements) that differentially affect firms and industries. The third component may, similarly, reflect purely transitory shocks to firm activity; alternatively, this component may be due to purely random measurement error (or ‘noise’) in the data.

In particular, to capture these univariate features, we assume that S and P satisfy the following statistical processes:

(1a)    

and

(1b)    

for firm-i in year-t, where (for j=P,S) represents a time-invariant firm-specific component, is a time-varying persistent component, and is a purely transitory component. We assume throughout this exercise that all shocks and other variables here are normally distributed. Noting that ^lP is simply the difference between S and P implies

(2)    

In terms of the autocorrelation patterns described above, the permanent component () provides a permanent correlation across different lags, the purely transitory component () contributes only contemporaneously, while the component () facilitates variation across correlation lags. Given the raw autocorrelations for S and P exhibit steady (geometric) decay after lag-1, we suppose that both S and P follow stationary first-order autoregressive, AR(1), processes:

(3a)    

and

(3b)    

where ρ^p and ρ^S are coefficients which capture the persistence in the AR(1) processes, and and are innovations. Furthermore, given the similarity of the autocorrelation patterns for S , P (and ^lP), we assume that ρ^p = ρ^S= ρ. A convenient consequence of this is that the second component of the ^lP process also follows the same AR(1) process – i.e. .

Second, and more substantively, to the extent that a firm’s GST sales and purchases capture and reflect relevant aspects of the firm’s production process, we would expect these measures to be closely related via the firm’s business activity. We incorporate this feature by allowing the respective components of S and P to be related through common business activity shocks, and we also assume that such business activity shocks are related via firms’ value-added or “mark-up” from purchases to sales.

In particular, first, we assume the permanent components of differences are due entirely to common business activity differences and related by

(4)    

where μ represents the firm’s relative value-added between purchases and sales. Second, we assume the sales and purchases’ innovations to the AR(1) components include a common business activity innovation, which is similarly affected by value-added between purchases to sales:

(5a)    

and

(5b)    

where α and β represent the relative contributions of the common business activity innovation to the AR(1) shocks of purchases and sales respectively. Finally, we similarly assume that the purely transitory shocks to S and P have common business activity components plus idiosyncratic components:

(6a)    

and

(6b)    

where γ and δ represent the relative contributions of the common business activity innovation to the AR(1) shocks of purchases and sales respectively.

Given this set-up for the model, we choose parameter values as follows. First, based on the result (see Table 4) that sales per hour are on average around 35 percent higher than purchases per hour, we adopt a relative value-added rate of μ =0.35. The AR(1) correlation coefficient ρ=0.7. A strong implication of stylised fact 2 is that, contemporaneously, S and P are highly correlated. For this reason, we set α,β, γ, and δ = 0.9, so that the components common to S and P ( and ) dominate the idiosyncratic components

All innovations, both common, and specific to, S and P respectively, including the permanent business activity component, λ_i, are normally distributed with mean zero and variance one. These innovations are randomly generated for one thousand hypothetical firms for twenty periods. They are then used to generate S , P and ^lP according to the model described above. To allow the autoregressive process to stabilise, only the last ten periods of synthetic data are used to produce autocorrelation charts similar to those in section 4.1.