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2017-11-02Macro `n Cheese

Doing the unthinkable

This week, our sometimes quixotic Roger Josefsson has set out to do the unthinkable and fight the long run Phillips curve! Does he succeed? Let us just say that he would probably have done better tilting at windmills.

If there is something sacred in economics, it is the assumption of the vertical long-run Phillips curve (LRPC). We have touched upon the Phillips curve in previous posts, but this time I thought we could utilize a new feature in Macrobond version 1.18; the vector error correction mechanism (VECM), to get a deeper understanding.

Mainstream macro models typically assume a stable or time-varying non-accelerating inflation rate of unemployment1 (NAIRU), both sharing the trait of a vertical LRPC where NAIRU is compatible with any level of inflation. Empirically, this implies that there should be no long run relationship between inflation and unemployment, an assumption often imposed without any testing despite the obvious possibility of, e.g., a reverse impact of unemployment on inflation.

Admittedly, some empirical work has been done on this subject, mainly confirming that “a credible optimizing policy maker results in a perverse positively-sloped Phillips curve” in the words of Haldane and Quah (1999). However, still consistent with a “sufficiently powerful” policy maker, other studies reach the conclusion that the LRPC is also downward sloping. Then again, others simply adhere to the view that there might be long-run trade-offs between output and inflation or, even, that there is (almost) no utilizable Phillips relationship at all. Admittedly, this latter research highlights another intriguing possibility, that of inflation being stationary instead of unemployment.

Here, I will, nonetheless, focus on the possible existence of an exploitable, non-stationary, long-run relationship between inflation and unemployment. Do note, also, that the unemployment rate is bound by its denominator and cannot indefinitely remain integrated of the first order; cannot indefinitely remain an I(1)-process. This means that we could expect to see different test results as the samples grow over time and underlines that our results are, at best, approximations of the full relationship.

If inflation and unemployment rate can be approximated as I(1)-processes it means that either inflation and unemployment do not cointegrate2, or; there do exist a cointegrating relationship, why the relationship contains additional statistical information that standard macro models refrain from exploiting!

First, let us do some good old eye ball econometric studies (I have chosen to examine the OECD GDP-deflators and OECD UNR:s for USA, Germany, France and Norway but it should be a fairly easy task to exchange the codes in the downloadable MB-document into your countries and variables of choice.


1A constant NAIRU could possibly be found in structurally stable economies with little nominal rigidities. If an economy exhibits structural breaks and have higher degrees of nominal rigidities; Sweden being a good example, a time-varying NAIRU is more probable.

2In which case the non-stationary component can be removed by purely statistical methods (such as, e.g., Hodrick-Prescott or Kalman filters).

Macrobond Document
Figure 1: Inflation rates (GDP-deflators from OECD:s database)
Macrobond Document
Figure 2: Unemployment rates (from OECD:s database)

With some effort we can detect some type of falling mean in inflation rates over longer horizons for almost all countries. As for unemployment rates there seems to be a quite clear rising mean over longer horizons for most countries, with the possible exception of the US. If this is correct than the series could be stationary around some deterministic trend.

We are not yet able to perform series by series unit root tests such as the widely used Augmented Dickey-Fuller (ADF) test in the MB-application. That said, the cointegration tests (I have chosen Johansen, as we have quite large samples) of course implies a test of unit roots, simultaneously controlling one series for cointegration with another, rendering such testing superfluous. In short, we are not in any way restricted to perform our cointegration tests and subsequent VECM-analysis by the “lack of” ADF, even if it would perhaps facilitate the model selection process.

In this context it should be noted that I have refrained from controlling for possible regime shifts etc. Such dummies could easily be added, or you could just simply adjust the sample windows. Also, here, I somewhat recklessly fall back on the notion that cointegration implies that any long-run relationship is “super-consistent” gradually reducing any bias from omitted variables.

The existence of a cointegrating relationship could, in theory, be said to diminish the need to check for robustness. However, at least for empirical purposes, it is still a necessary discussion why I have added and subtracted the usual exogenous measures of supply and demand shocks (energy, IMPI, Productivity, FX) from my models. For the rigid economies France, Germany and Norway we detect quite robust coefficients in the long-run (cointegrating) relation3. This is not the case for the US economy, but could perhaps also be expected given the stability and flexibility of the US economy4. Admittedly, our cointegration rank tests in many cases point to our US VECM model(s) being borderline full rank (meaning variables are close to stationary and that we could use a traditional VAR in levels) implying that we should be extra careful when interpreting the results for the US economy.

Macrobond Document
Figure 3: The cointegrating relationships

Note: The cointegrating relationships have, for ease of use, been expressed in ordinary terms, whereas testing and modelling has been performed in log diffs. ‘π’ denotes inflation (GDP-deflator) in annualized terms and ‘u’ is the unemployment rate.

Source: Macrobond

As most ECM:s, our models are inclined to treat spikes as being “structural”, which has resulted in very weak forecasting abilities in the period surrounding/following the global financial crisis of 2007/08. That said, the intention here is not to necessarily create a forecasting tool, but rather to discuss a feature of standard macro models.

I also tried to adjust starting periods, which seemed to have some positive impact (possibly due to regime shifts or other shocks) on the performance of the models, why I have adjusted the sample periods according to the most recent stabilization political set-up; usually some type of monetary policy regime with attention on inflation developments, if not always a fully-fledged inflation targeting regime. Hence, for the US, I have chosen the start of the great moderation (c. 1985), for Germany the strategic monetary policy change of the BuBa (c. 1975) and for France and Norway, the post-ECU periods (c. 1995). An alternative would, e.g., be to try dummies for different regimes, but that would probably be an over-kill (and sample-sizes are not an issue) for the purposes of this blog post.


3Since these economies have experienced quite a few structural shocks and “regime shifts” (reunification, crisis, EU/Euro cooperation etc) over the past few decades this is still somewhat surprising, despite their “rigidness”.

4But it rhymes well with our eye-ball econometric findings.

Macrobond Document
Figure 4: Even for the US, a Phillips VECM approach works! Albeit just barely.
Macrobond Document
Figure 5: Germany exhibits a stable long run relationship between inflation and unemployment.
Macrobond Document
Figure 6: France also exhibits a stable long run relationship between inflation and unemployment.
Macrobond Document
Figure 7: ...And yes, in Norway too.

As I set out to discuss the assumption of a vertical LRPC we can’t really stop at just showing that it is possible to identify and utilize a long-run (no matter heretical) relationship between inflation and unemployment. Unexpected as this relationship may be, it is not by itself sufficient to invalidate the existence of a vertical LRPC. To delve deeper into that issue we will have to utilize the statistical tools and reports created by the Macrobond application. In the following, I will use Germany as the main example and refer to the results for the other countries only when necessary.

Macrobond Document
Figure 8: Cointegration rank tests for Germany.

As can be seen, we can easily reject the hypothesis of no (‘0’) cointegrating relationship. As I used the default trace test as criteria instead of Maximum Eigenvalue the application highlights that the trace criteria cannot reject the hypothesis of a single (‘1’) cointegrating relationship (if I had additional endogenous variables in the model there would of course be tests for further relationships).

I have also chosen an automated selection process which (for Germany) results in two lags on the endogenous no matter what information criteria and settings I choose (in the MB-document I have applied the Schwarz information criteria and default settings of maximum two lags on the endogenous, maximum two regressors and maximum one lag on any exogenous variables).

The view that there is indeed a long run relationship between inflation and unemployment is underlined, and any initial fears that inflation might not be an I(1)-process5 are put to rest by the fact that the unemployment rate is highly significant in the cointegration relation (table below). The interpretation being that the inflation rate alone is not stationary.

Macrobond Document
Figure 9: A screen-shot full of information

Nonetheless, the signs on the loadings show that inflation is doing all the error correcting work in our model as it has the expected negative sign whereas the loading sign of the unemployment rate is positive (circled red), implying it is de facto increasing the error rather than correcting it6 (usually they should lie between 0 and -1). This goes hand in hand with Haldane and Quah’s quote on the “perverse” positive slope of the L-R Phillips curve above and suggests that something else is driving unemployment.


5I.e., for those of you still looking for an ADF-test.

6This is probably also why the model reports moduli ≥1 which indicates it is unstable. Admittedly, this also suggests “hidden cointegration” and that the single cointegrating relation is not really sufficient to explain long-run corrections of my system and a host of related problems that a more serious attempt should try to take into consideration.

Macrobond Document

Note: t-values within parenthesis. Usual denotations apply.

Figure 10: The model for Germany in a more customary form.

However, to understand what then affects the unemployment rate in the long run it is fruitful to run a couple of impulse-response analysis and compare the bivariate model to one expanded with different supply and demand shocks (e.g. those mentioned above). However, this would probably also necessitate identifying assumptions, or tests of if the residual covariance matrix could be treated as diagonal7.

This was/is no easy task. Not only do we have strong data restrictions but suddenly it seemed as if “structural breaks” (the reunification and the introduction of the Euro comes to mind) were rampant. Only with strong modifications of the model was I able to compute some type of non-explosive IRF:s, but such an amount of data mining can best be labelled “unadvisable”. Anyhow, whatever it was that I concocted, it gave rise to analogous patterns in the bivariate as well as the “extended” models8 . The initial “perverse” positive reaction seemed to be fortified, and the similarity between bivariate and extended model specifications suggests that the observed long-run negative relationship between unemployment and inflation is due to persistent shocks form unemployment. Plainly speaking, this implies that the source of the non-stationarity of both variables is the real side of the economy (represented by the unemployment rate in our bivariate model).

To conclude, what initially seemed like a rejection of the oft assumed vertical LRPC turned out, if anything, to be a confirmation of the canonical New-Keynesian LRPC, where real, micro-founded, variables determine inflation. The positive co-movement in inflation and unemployment for Germany (by and large this is also recognized by our model of France) is probably a testament to unaccounted for exogenous factor(s) which are picked up by the rise (persistent shocks) of unemployment in our model(s).

So, no, my challenge of the vertical long-run Phillips curve was by and large (and perhaps thankfully) unsuccessful. However, the results do suggest that an empirically identifiable constant NAIRU is hard to come by for most countries (at least for Germany, France and Norway). Also, as cointegration was apparent (again mainly for Germany, France and Norway), special care should be taken to explore any long-run relationship. Arbitrarily using time-varying statistical filtering techniques that many researchers seem fond of risk throwing away important and exploitable information.


7I have not formally tested it, N.B., albeit the near ‘0’ diagonal values suggest it is possible.

8I strongly advise the interested reader to perform the specification of the model and this particular analytical step in a considerably more coherent and ambitious way.



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