*Update November 20, 2014:* English translation of new Ekonomistas post.

A closer look reveals that the different results of Anderson and Jonung are due to a trivial econometric error.

Lars Jonung has repeatedly in Swedish media in sweeping terms criticized my estimate of the increase in average unemployment caused by the fact that average inflation in Sweden significantly undershot the target during 1997-2011. Every time I have replied and shown and pointed out that he is wrong on all counts, lately in this Ekonomistas post (in Swedish) and in Svenska Dagbladet (also in Swedish). Now, Fredrik Andersson and Jonung have written a long paper with a series of regressions which are supposed to show that my results are not robust. The idea of Andersson and Jonung is, despite the common practice in similar studies and despite econometric problems with overlapping data, to use annual inflation instead of quarterly inflation. However, on closer inspection, their regressions turn out to be misspecified and not reliable. When I examine corrected versions and run new regressions with annual inflation, they confirm my previous estimate. Again it seems that Jonung is wrong on all counts.

Jonung has, despite sweeping claims, not been able to explain what could be wrong in all the robustness tests that my estimate is exposed to in my paper and that confirm the original result. My result holds not only for CPI inflation but also for inflation as measured by the CPIX/CPIF price index or by the GDP deflator. It holds regardless of whether inflation expectations are included as explanatory variables or not. The calculation involves estimating a short-run and a long-run Phillips curve. The key is the estimate of the slope of the long-run Phillips curve. In accordance with the established econometric standard, I use data for quarterly inflation (the percentage change in the price level from the previous to the current quarter).

### Overlapping data or not?

An alternative to using quarterly inflation is to use annual inflation (that is, the percentage change in the price level over the last four quarters). But as the annual inflation rate is a moving average of four quarterly inflation rates, this means using overlapping data. This leads to some econometric problems and less reliable estimates (see footnote 16 of my paper). There are no advantages to using overlapping data, and there are econometric reasons to avoid overlapping data when possible, as explained in a paper by Harri and Brorsen. In my case, I actually started to estimate the Phillips curve using annual inflation, but was advised by Bertil Holmlund, Professor Emeritus at the University of Uppsala and a specialist in labor market economics and labor econometrics, to follow the established standard and use quarterly inflation. In my case, the estimate of the slope of the Phillips curve is about the same with quarterly inflation as with annual inflation, but the fit to the data and the precision of the estimate is much better with quarterly inflation.

Given the econometric problems with overlapping data, if regressions with annual inflation give different results from regressions with quarterly inflation, this does not mean that the latter would be unreliable. Rather it suggests that the regressions with the former are unreliable.

In spite of the econometric reasons to avoid overlapping data, Andersson and Jonung choose to use annual inflation in their regressions. The reason is supposed to be that because inflation expectations refer to annual inflation, then annual inflation should be used in the equations instead of quarterly inflation. But this not a valid reason. First, it is normal and established in the literature to use expectations of inflation further into the future or over a longer period than that over which inflation is measured (see, for instance, this paper by Jeffrey Fuhrer). In the Swedish case this may be especially justified by central wage negotiations often resulting in multi-year agreements, for which the social partners’ expectations of and assumptions about inflation over the whole period of the contract would be relevant. Second, using expectations of annual inflation in conjunction with quarterly inflation can be seen as making the implicit and natural assumption that expectations of inflation over the next year is fairly uniformly distributed over the four quarters of the year, that is, inflation for each quarter is expected to be approximately equal to the average over the year.

If one nevertheless insists on using annual inflation, in spite of the estimates being less reliable, one has to be careful. The dynamics when a number of moving averages are included in the regression becomes somewhat complex and must be carefully examined. Andersson and Jonung did not do that in their paper, and they did not discover that their regressions and equations are misspecified.

### Misspecified regressions

Andersson and Jonung run a regression corresponding to equation (1) in the table below (equation (1) in their table 3), with annual inflation as the dependent variable and the average unemployment rate over four quarters lagged one quarter (UBAR(-1)) and the change in unemployment during the four quarters (U-U(-4)) as explanatory variables. (Click for a larger table in a separate window.) The coefficient of the average unemployment rate corresponds in this case to the slope of the long-run Phillips curve (the row for UBAR(-1), highlighted in yellow in the table). It is (minus) 0.64, slightly less steep than my benchmark estimate of the slope, 0.75. A less steep long-run Phillips curve implies that the average increase in the unemployment rate, compared to if inflation had been on target, gets bigger. Given that average inflation undershot the target by 0.6 percentage points, the average increase in the unemployment rate will be 0.6/0.644 = 0.93 percentage points. This is slightly larger than, but not significantly different from the increase of 0.6/0.75 = 0.8 percentage points in my benchmark estimate. The table also separately shows the slope and the corresponding average increase in the unemployment rate (also highlighted in yellow).

Jonung has erroneously repeated several times that my estimate would not be robust to including inflation expectations among the explanatory variables. But my paper (and this blog post, in Swedish) clearly shows that the estimate is valid regardless of whether inflation expectations are included or not, when quarterly inflation is used. But what if annual inflation is used, as Andersson and Jonung prefer? In equation (2) expectations held one year ago of inflation one year ahead are included among the explanatory variables. This corresponds to a so-called New Classical Phillips curve, which according to how Swedish wage-setting is done should be the most relevant variant of the Phillips curve for Sweden. The coefficient of inflation expectations is not significant. The slope of the Phillips curve is 0.68 and the increase in the average unemployment rate is 0.88, about the same as in equation (1). In equation (3), instead inflation expectations held the previous quarter of inflation one year ahead are included. This corresponds to a so-called New Keynesian Phillips curve, which given how Swedish wage setting is done should be less relevant. The coefficient is significant and close to 1. The slope of the long-run Phillips curve for given inflation expectations is lower, 0.28. and the increase in the average unemployment rate is as high as 2.1 percentage points.

However, equations (1)-(3), and all equations with annual inflation that Andersson and Jonung estimate, are misspecified. The low Durbin-Watson statistic might be seen as a warning signal. If Andersson and Jonung had done some robustness tests, they might have discovered that the dynamics of the equation, with all these overlapping data, is more complicated than they have assumed. This is shown in equation (4), which is the same as equation (1) but with the annual inflation rate lagged one and two quarters included among the explanatory variables. We see that these lagged inflation rates have considerable explanatory value, with very significant coefficients. We also see that the adjusted R^{2} is a good deal higher, 0.83 against 0.46 in equation (1). The slope of the Phillips curve now has to take into account the lagged inflation terms. The slope is therefore given by 0.210 / (1-1.088 + 0.498) and equals 0.59. The increase in the average unemployment rate is then 1.0 percentage points.

Furthermore, there is no reason to take for granted the specification of Andersson and Jonung of how the unemployment variables enter, that is, as a lagged four-quarter moving average of the unemployment rate (UBAR(-1) and the four-quarter change of the unemployment rate ( (U – U(-4)). For simplicity, I keep their specification here, but in a more careful analysis it is necessary to examine this more closely and let them be determined by the data.

Equations (5) and (6) are the same as equations as (2) and (3) but with the lagged annual inflation rates as explanatory variables. For equation (5), the slope of the Phillips curve is 0.68 and the increase in the average unemployment rate is 0.88 percentage points. For equation (6), the slope is 0.41 and the increase in the average unemployment rate is 1.5 percentage points.

In the paper, I show that the results also hold if, instead of CPI inflation, one uses quarterly inflation measured by the CPIX/CPIF price index (CPIX inflation until 2008Q1, when CPIX was abandoned, and CPIF inflation from 2008Q2, when CPIF was introduced) or by the GDP deflator. Equations (7) and (8) show the results when annual inflation is used instead. For these cases, the results for the slope of the Phillips curve and the increase in average unemployment are very similar to those I get with quarterly inflation, but the precision is lower.

### My result holds also when annual inflation is used

Equations (1)-(3), as well as all of Andersson and Jonung’s equations with annual inflation, are thus misspecified and their results are not reliable. When the misspecification is corrected, the regressions with annual inflation give somewhat larger (but not significantly different) increases in the average unemployment rate than my benchmark estimate, 0.8 percentage point, when quarterly inflation is used.

Furthermore, with annual inflation, the reliability of the results is lower and the uncertainty in the estimates is larger. The fact that the expression for the slope of the long-run Phillips curve contains uncertain estimates of the coefficients of lagged inflation in the denominator means that the confidence interval for the slope, and thus the confidence interval of the increase in average unemployment rate, gets bigger.

This illustrates the benefits of using quarterly inflation. In this case, the equation that fits the data becomes simpler, and the precision of the estimate of the slope and the average increase in the unemployment rate is better.

Overall, this exercise thus shows that my results are robust even for the worse option of using annual inflation. My estimate using quarterly inflation, that the average unemployment rate has become about 0.8 percentage points higher (corresponding to about 38,000 jobs) during 1997-2011, is as a more precise and somewhat lower estimate compared to that based on annual inflation. Again, Jonung is wrong on all counts. Time to end the debate?