4 Time-series properties and transition rates
This section presents time series analysis of firm labour productivity and its components. The aim is to gain a better understanding of the variability and noise present in the data and the persistence of firms’ labour productivity in particular. We describe the patterns using two metrics. First, Section 4.1 presents the autocorrelations of, and cross-autocorrelations between, sales per hour, purchases per hour and labour productivity. Second, Section 4.2 describes the 1-year, 4-year and 9-year transition probabilities between quartiles (and also entry and exit status) for each of sales per hour, purchases per hour and labour productivity. The group of firms we analyse is all firms in our sample, regardless of their industry or length of time in operation.
4.1 Autocorrelations
Simple measures of the association between the sales per hour, purchases per hour, total hours and labour productivity are developed by estimating the correlation between these variables and by estimating the association of a variable with its own values in previous and future time periods using autocorrelation analysis.
Autocorrelation patterns for sales per hour, purchases per hour, total hours and labour productivity are shown in Figure 1. The centre line shows the average autocorrelation for various lag lengths (where lags are in years). With ten years of data, one for each of 1994 to 2003, there are nine one period autocorrelations from which to derive an average, one for each pair of consecutive years. Similarly there are eight two period autocorrelations and so on. The top line shows the maximum autocorrelation for each lag length. The bottom line shows the minimum autocorrelation for each lag length. Obviously the three lines meet at the same point for the nine period autocorrelation (lag 9 years).
From Panel A in Figure 1 it is apparent that the relationship between a firm’s sales per hour in different periods becomes steadily weaker as the length of time between periods increases. For example, the average autocorrelation between sales per hour in consecutive periods is around 0.65 compared to 0.25 for the nine period autocorrelation. Autocorrelations for the same lag length can vary considerably. The highest autocorrelation between consecutive years was 0.87 and occurred between 2002 and 2003 while the lowest was 0.32 and occurred between 2000 and 2001.
The autocorrelation pattern for purchases per hour is shown in panel B of Figure 1. The picture looks very similar to that for sales per hour. The relationship between a firm’s purchases per hour in different periods becomes steadily weaker as the length of time between periods increases, tending to about 0.25 by the seven year lag.
Panel C of Figure 1 shows the autocorrelation pattern for firms’ total hours. The labour input of firms appears very persistent. The average autocorrelation between firms’ total hours in consecutive periods is around 0.98 and is still more than 0.91 for the nine period autocorrelation. This suggests that labour hoarding is quite common and that rather than vary labour inputs in response to variations in demand they tend to compensate in other ways, for example through changes in inventories and variations in the rate of capital utilisation and therefore labour productivity. It may also partly reflect the extent to which observations for this variable are imputed, approximately 22 percent.
- Figure 1 Autocorrelations for sales per hour, purchases per hour, total hours and labour productivity
Panel D of Figure 1 shows the autocorrelation pattern for labour productivity. The pattern here differs from those of sales per hour and purchases per hour. The relationship between a firm’s labour productivity in different periods is not as strong and weakens more rapidly as the length of time between period’s increases. It is interesting that sales per hour of a firm, an alternative measure of labour productivity, is less volatile than the value-added measure of labour productivity.
Cross-autocorrelation patterns are shown in Figure 2. The cross-autocorrelation pattern for sales per hour and purchases per hour is shown in panel A. The leftward part of the chart shows the cross-autocorrelations between sales per hour in year t and purchases per hour in years prior to year t. The rightward part of the chart shows the cross-autocorrelations between sales per hour in year t and purchases per hour in future years. The two parts are quite symmetrical. Given the potential problem mentioned in Section 2 concerning the appropriate timing of sales and purchases when constructing value-added, it is interesting to note that the highest correlation between sales per hour and purchases per hour is the contemporaneous one.
The cross-autocorrelation pattern for sales per hour and labour productivity is shown in panel B. It looks very different to that for sales per hour and purchases per hour. The relationship is very weak, even contemporaneously. This further reinforces the point that the choice of measure for labour productivity may materially impact on results.
The cross-autocorrelation pattern for purchases per hour and labour productivity is shown in Panel C of Figure 2. The relationship looks to be even weaker than that for sales per hour and labour productivity. In some cases the correlation between purchases per hour and labour productivity is even negative.
Because of firm entry and exit, the sample of firms we have used to calculate the autocorrelations and cross-autocorrelations presented in this section are not constant over the sample period. This may have had some influence over the patterns we observe. To check for this possibility, we produced the same autocorrelations and cross-autocorrelations for the group of firms that were in operation in every year between 1994 and 2003. The autocorrelation patterns for this group of firms are similar to those presented in this section for the full sample of firms. We also compared autocorrelation patterns across different industries and, although the patterns did vary across industries, these differences did not appear to be systematic. For example, we were unable to discern systematic differences between industries that were likely to be capital intensive and those that were labour intensive, or between industries that exhibited high rates of entry and exit compared to industries that exhibited low rates of entry and exit.
