The empirics of granular origins: some challenges and solutions with an application to the UK

Abstract

We study the effects of firm-level microeconomic fluctuations on aggregate productivity in the United Kingdom. We show that a standard measure of residual productivity growth of the largest UK firms (the ‘granular residual’) produces results that are partly counter-intuitive and statistically insignificant. To combat this, we propose a refinement to the widely used control function approach to estimating technology shocks in a production function, which is aimed at accounting for firm-level heterogeneity and the potential existence of common shocks. Using this approach, we find that idiosyncratic firm-level shocks matter for the UK; the ‘granular residual’ can explain around 30% of aggregate UK productivity dynamics. We also show that simplifications of our approach, which do not control for firm-level heterogeneity and the existence of common shocks, do not perform well empirically, highlighting the importance of identifying firm-specific shocks correctly in order to properly test the ‘granularity hypothesis’.

Appendices

Appendix A. Simulation exercise

We conduct a simulation exercise in a hypothetical economy consisting of N = 100,000 firms³², where we then randomly draw the employment shares of the firms from a Zipf distribution to conform to the power law requirement of the GR discussed in the main text. We allow the firms to be hit by random idiosyncratic and common productivity shocks (described below in more detail), and calculate the GR measure for the largest 100 firms based on Models 1 and 5 in the main text. Finally, the aggregate whole–economy productivity growth rate is then regressed on the different GR measures to estimate the significance of the GR coefficients and the (adjusted) R² of the regressions. In the results shown below, we assume that there are L = 1000 loops of the simulation, and each loop consists of T = 100 time periods.

We want to match the definitions of the shock processes as closely as we can with models 1 and 5 introduced in the main text. First, to simulate a world where there are no idiosyncratic shocks, we assume that firm-level productivity (or TFP) ω_it follows an AR(1) process of the following form:

$${\omega }_{it}={\rho }^{s}{\omega }_{i,t-1}+\beta {\mu }_{t}+{c}_{i},$$

(A1)

where (unlike in the empirical models we use in the main text) we assume that ρ = 1 for simplicity. There is also a shock common to all firms, μ_t. β is the coefficient on the common shock, which is here allowed to be random, rather than 1.³³ We also allow for a firm fixed–effect c_i, although in practice, this has very little effect on the results. The common shock is drawn from a N(1, 1) distribution, which implies an average growth rate of the economy of 1%. The intuition of the model in equation (A1) is that since there are no idiosyncratic shocks, a correctly estimated GR should be zero.

Second, we simulate a more general version of the model, but now allowing for idiosyncratic, firm–specific shocks χ_it:

$${\omega }_{it}={\rho }^{s}{\omega }_{i,t-1}+\beta {\mu }_{t}+{c}_{i}+{\chi }_{it},$$

(A2)

We also allow for two versions of this model; one where the idiosyncratic shocks are drawn from a N(0, 1) distribution, and one where they are drawn from the same distribution, but the size of each shock is divided by 10. We do this because we want to examine what the effect of the relative size of the idiosyncratic vs the common shock is on the significance of the GR.

To calculate aggregate productivity in this example, we again use the Melitz and Polanec (2015) definition at time t:

$${{{\Phi }}}_{t}=\mathop{\sum}\limits_{i}{s}_{it}{\phi }_{it},$$

(A3)

where the employment shares s_it ≥ 0 sum to 1 and ϕ_it denotes the log of firm i’s productivity at time t. For simplicity, we will assume that the firm employment (s_i) stays constant over time.

With these simulated results, we then calculate the GR for the largest 100 firms for each time period, using Model 1 for equation (A1), and Model 5 for equation (A2). Finally, we estimate an OLS regression that has the following form, for each of the 1,000 loops (and where the sample size is T = 100 for each loop):

$$AL{P}_{t}=c+G{R}_{t}+G{R}_{t-1}+G{R}_{t-2}+{e}_{t}$$

(A4)

where ALP_t is aggregate productivity growth at time t, c is a constant, GR_t is the relevant GR measure and e_t is an i.i.d. error term.

Figures 10 and 11 show the main results in terms of the estimated adjusted R² for each of the 1,000 loops of the simulation, for the three types of shock processes. There are a number of points worth highlighting:

• First, the exact shape of the Zipf law distribution can cause significant volatility in all the GR estimates, as suggested by the fact that the GR estimates are much more volatile in Fig. 10 (which draws the firm sizes again for each loop, though not for each t) than in Fig. 11 (which keeps the firm sizes fixed so that the share of the largest 100 firms is around 0.3, which is the average share in Fig. 10).

• Second, even though we know the the GR for the case with the common shock only should be zero by construction, Model 1 shows a significantly positive GR throughout the 1,000 loops (with an average of 0.32 for the case with random firm sizes, and 0.22 for the fixed firm sizes).

• Third, Model 5 detects a significantly positive GR (with an average of 0.23 for the case with random firm sizes, and 0.22 for the fixed firm sizes) for the idiosyncratic shock, when this shock is relatively variable compared to the common shock, but detects a GR that is effectively 0, when the shock is relatively less variable – just as would be expected based on the definition of Model 5.

Fig. 10

Simulation results for adjusted R² – random firm size

Fig. 11

Simulation results for adjusted R² – fixed firm size

Appendix B. Data

Data used in various parts of our analysis comes from three data sources: Refinitiv Worldscope, Office for National Statistics (ONS) Business Structure Database (BSD), and OECD/ONS labour productivity data. This appendix describes the data we use from these datasets in detail.

Worldscope from Refinitiv is a proprietary dataset that includes financial account information on large (mainly listed) UK firms since the 1980s. For our analysis, we use data on the following variables:

• Turnover for the largest firms for each year. Where data on international turnover exists, we compute domestic turnover as the difference between total and international turnover. This is more comparable with how national aggregates are constructed, given the nature of the UK as a small, open economy.

• Employment, used as the denominator in the firm-level labour productivity calculation. For those firms that report international sales, we assume—given lack of data—that the share of domestic employment in total employment is the same as the share of domestic turnover in total turnover. This will affect the employment weights of the individual firms, where applicable. However, this results in productivity (turnover divided by employment) being the same, whether measured by total turnover and employment or by their domestic counterparts. Given data limitations, we think this is a reasonable (and only available) choice.

• Cost of goods sold, used as a measure of variable costs in the control function approach to estimating production functions.

• Net property, plant and equipment as a measure of firm–level capital.

Turnover, cost of goods sold and firm–level capital are deflated with the aggregate ONS GDP and gross fixed capital formation deflators.

Importantly, we have noticed a number of mistakes in the Worldscope data on employment and international sales (and to a lesser extent, cost of goods sold). Intuitively, employment and international sales are not financial account items, which likely explains the presence of errors in the recording of it in the Worldscope dataset. While correcting for all mistakes in the entire data would not be feasible, we have gone through the largest errors for those firms that contribute most to the dynamics of the GR measures and replaced the erroneous observations with the correct ones from the annual reports of the firms in question, when these have been available. In particular, the number of corrections we do are the following:

• Cost of good sold; corrections for 10 firms.

• International sales; corrections for 70 firms.

• Employment; corrections for 30 firms.

It is worth emphasising that after these corrections, we consider the data to be of good quality for the analysis we are carrying out. There is no reason to doubt the vast majority of data on turnover, cost of goods sold and firm–level capital, as these are recorded in firms’ financial accounts. The employment data is of lower quality, but we do not rely on it for our main results in any case.

BSD covers the universe of all UK enterprises, based on an annual snapshot from the Inter-Departmental Business Register (IDBR). On average, there are over 2 million enterprises annually in the BSD. In the IDBR, turnover is updated via administrative sources (HMRC VAT and PAYE records) and ONS Business Surveys. We use enterprise level turnover for calculating the power law distributions, based on the BSD vintage of 2017.

To calculate the productivity contributions of the largest firms for the Melitz and Polanec (2015) methodology, we use the ONS ARD database. This database includes firm-level financial account data based on the Annual Business Inquiry (ABI), Annual Business Survey (ABS) and employment data based on the Business Register and Employment Survey (BRES). We use reporting unit level turnover and employment data to calculate estimates of turnover productivity. Note that the data is only available for 2004–2018, with methodological breaks during this period, so the results should only be seen as indicative and motivating the main analysis using the Worldscope data. Importantly, we cannot use this dataset in our proposed control-function approach as there exists no data on fixed capital in the ARD database.

ONS labour productivity data is used for the regressions reported in Tables 3–5, where aggregate labour productivity (output per head) is the independent variable. In addition to this, Fig. 1 uses OECD data on hourly labour productivity for selected economies. All the ONS and OECD aggregate data is publicly available.

Table 3 Predictive power of the first-order granular residual for UK aggregate productivity growth

Table 4 Predictive power of the second-order granular residual for UK aggregate productivity growth

Table 5 Predictive power of the granular residual for UK productivity growth – robustness checks