My intuition would tell me that the inflation variables and GDP growth (better estimated as the first differences of price indices and real GDP volume are stationary which your tests do not appear to reject. An interest rate is a rate of return and I would not be inclined to use its log in my analysis. I would be inclined to use the log of the oil price. Thus the variables that I would tend to include in my VECM analysis would be

• log of prices ( integrated inflation - in the VECM the independent variable would be the log difference of price or inflation)
• real gdp
• interest rates (?)
• log exchange rates
• log oil price

it is almost impossible to trace through a program such as yours with out the data and I a not going to invent some to check what is going on

Hello John,

Thanks a lot. I will try this. I just succeeded in running this model (identifying a normal result of the AIC and BIC) by using the log of Real instead of nominal oil prices, Interest rate (federal funds rate), core inflation changes in percent, and unemployment rate...i slightly changed the approach by starting from the phillips curve ...

Now i just wonder whether a vecm does not require to use all essential variables (so skip variables for sake of testing a theory).

Information criteria can give ambiguous results for model selection. However, the fundamental criteria is for the model to satisfy the underlying assumptions including no autocorrelation, no heteroscedasticity, no normality, no non-linearity parameter constancy etc. With time series data, autocorrelation is typically the main problem so this should be tested and, if you have a large sample, you can use heteroscedasticty consistent coefficient standard errors and appeal to asymptotic normality to deal with any evident heteroscedasticity and non-normal residuals, respectively. I would say that when information criteria are ambiguous you should choose the minimum lag length that overcomes model misspecification. That would, at the very least, mean rejecting significant autocorrelation (for the whole system, system tests are available in software) although you may want to check tests for other forms of misspecification.

You have about 70 observations and 5 variables so that each lag costs 5^2 = 25 parameters, if I understand correctly. So you can afford only 2 lags to avoid a negative number of degrees of freedom. That's why you get -Inf. (infinity) or NaN results, I fear.

I think John Frain is correct. You may be over-differencing real gdp and the inflation rate, causing a misspecification of the VECM. Another thing in model specification is that you should be guided by some underlying theory in deciding the included variables, rather than trying combination of variables in hope of obtaining some sensible results. Good luck with your research.

Your specifiction seems to be rather accidential, at least, I do not see a clear theory behind it. Obviously, you mix different transformations of the variables (percentage(?) changes, absolute logarithmic values), which, I think, cannot fit together. You should more carefully think over a good specification, i.e. developing an equation (function) showing how you or a theory you prefer core/healine inflation is influenced by other variables.

What concerns the lags, even a lag of 8 (i.e. 4 years) seems too long, 10 and 12 are, certainly, far from realistic. Shorter lags could also help you to come araound the degrees-of-freedom problems, Guy has mentionned.

Philipp, your question

Now I just wonder whether a vecm does not require to use all essential variables (so skip variables for sake of testing a theory).

The problem is that if you leave out variables the effect of the omitted variables may be reflected in the coefficient(s) of the included variable(s). As Anton Rainer has said you a clear theory before you do any econometrics This is particularly important in VECM analysis. The economic constraints that you need to identify your cointegration equations must be based on this theory.

If you do leave out variables you should supply a strong econometric argument that they can be excluded. You also need to show that your VECM is not misspecified paying particular attention to autocorrelation and coefficient stability using recursive estimation.

If you are worried about R programming have a look at gretl. It is free, easy to use, and does a lot of what you are trying to do. 74 observations is not a lot for elaborate analysis. I recommend Juselius (2007), The Cointegrated VAR Model, OUP. It is difficult but very comprehensive.

Anton Rainer So it is not possible to mix different transformations of the variables and anticipate this in the interpretation of the model later on? I have seen a lot of papers where variables are of mixed transformation.... but you would say e.g. I shall only combine growth rates OR transformations of absolute values together? Anton Rainer

You can, of course, mix transformations, but only if you have good (theoretical) reasons to do so.

Hello,

I based my model now on the four possible effects (transmission channels) described in Peersman, Gert; Van Robays, Ine

Oil and the Euro area economy

Economic Policy, Volume 24, Issue 60.

That is to say

1. direct effects (via the energy component in PCE inflation on core PCE),

2. the indirect/cost effects (the author uses Import and GDP deflator; I used Production Price Index),

3. the second round effects (the author used the price-wage ratio, nominal labour costs per employee, etc.... and I used the wage growth tracker of the Atlanta FED) and

4. The demand effects - I used the interest rate and the real GDP growth (the authors included also consumption and investments)

I wanted to keep my model relatively simple and so I replaced some variables or did not include others. When I enter mx-lag = 6, AIC shows me good results of 4 lags. My question is therefore:

1. Is this basic model appropriate and concluding in relation to the theory: XR_PCE_CORE

The direct effect on CPI of the oil price is, in effect, mainly an indirect (because crude oil is only consumed via fuels and other energy). But you could differentiate the effects by the lags. One can, e.g., assume and test, that the lag between oil price and the prices for petrol and diesel is near to zero. For heating oil, it is likely so, but the effect on households' consumption expenditure will have a higher lag.

Therefore, I would start with the analysis, how strong and quick fuel prices react to oil prices. The other effects, I would study them on a 1:1 basis, too, before building a big model. You should also look at the correlations between the independent variables. That could help you to find out lags between them. An estimation with several interdependent variables with lags up to 6 (i.e. 3 years) will likely give you spurious results.

Hello.

I constructed a model but I wonder if the ECT values can be considered as correct. I always thought they all, may they be significant or not, shall be negative and between -1 and 0. I have other results and I watched many clips on youtube where econometricians accept (do not recognize it) the fact that the coefficients are positive. In my view this would mean that the long term curves distance from each other instead of approaching? Do I have to change my model when some ECT coefficients are positive? Screenshot attached. Thank you very much in advance!

The idea that the coefficients on the coefficients on the ecm terms in a vecm should be between -1 and 0 is totally false. I challenge anyone to find an econometrics textbook that shows this.

In a vecm, the system will return to equilibrium if the absolute values of the eigenvalues of the estimated dynamic system are 1 (corresponding to a cointegrating relationship) or less than 1. (Your software may calculate the roots of the companion matrix which are the reciprocals of these eigenvalues - check your software manual.)

If you have a single equation with the ecm as the only explanatory term (no lagged first differences) then the system will return to equilibrium if the coefficient on the error correction term is between -2 and 0. If the system has lagged first differences the conditions on the error correction term depend on the value of the coefficient on the lagged first differences.

If you have more than one equation the coefficient on the ecm in one of the equations may appear to be 'explosive' but the values in the other equations may cancel that effect

74 observations are small for a 6 variable vecm. If your 't-statistics' are simply the value of the coefficient on the ecm divided by an estimate of the standard error these 't-statistics' do not follow a standard t-distribution. Thus the significance of the estimates is likely to be exaggerated.

Why do you want to estimate an ecm? You should start with much simpler estimations: e.g. oil price==>price of fuels==>prices of other products==>second-round-effect on product prices==>wages' increase etc. In any of these steps, you may also include other explaining variables like import prices and demand (GDP?)

You should also look-up the weights of the products/services (esp. fuels) for the CPI-calculations.

A problem is that many of these effects are likely rather short-term (wage effect will be longer, but dependent on how wage negotiations are handled in different countries.

If you look at prices in shops and supermarkets, you often find amounts like 3,99 or 299. One can therefore expect, that prices will not be changed for low changes in cost (or demand?), but they could overreact, if cost increase is high. There is likely an asymmetric reaction to increases or decreases. Therefore, it may be difficult to find-out a clear lag-structure. I am not sure, whether ecm is a good help in this respect, but it is easier to do for less explaining variables.

Anton Rainer Thank you for your answer, time and the idea. You are a great help to me and I appreciate your deep understanding and experience.

So which model would you suggest - an SVAR or an other model? I mean to model oil price==>price of fuels==>prices of other products==>second-round-effect on product prices==>wages' increase I need a clear hierarchy of shocks. I saw a very good SVAR approach of Lutz Kilian with gasoline --> headline --> core --> inflation expectations. But is this possible to modell in a VECM? I do not master SVAR in R but of course i would try.

I had the impression a VECM does not bring about satisfying results; at least i thought so until you told me the ECT coefficients do not have to be between 0 and -1. Now maybe it could work. I built a model based on a pass-through mechanism including: Headline inflation, output gap (supply channel), interest rate (interest rate channel), state expenditures (demand channel), import price index (imported inflation) and money supply.

You need to translate your links into equations. These equations will have unknown parameters. You then test if your data and model are consistent. If they are consistent then estimate the model. How you estimated the model depends on your model and data. Until one sees your equations and data one can not say how to estimate your model. The validity of VARs, VECMs, ECM's OLS, etc. depends on the model that you have built. You can not make causal inferences if you just put a data set into an econometric procedure an accept what comes out as causal. Such results will be spurious.

Anton Rainer has specified various facets of the model that you might build.

Lutz Kilian, A Comparison of the Effects of Exogenous Oil Supply Shocks on Output and Inflation in the G7 Countries, Journal of the European Economic Association Vol. 6, No. 1 (Mar., 2008), pp. 78-121. contains a relatively simple model. You might be able to use some variation on his model.

Like Kilian, you should probably be separating oil shocks into exogenous and endogenous. This is discussed in Kilian's article and in the references in the article.

As John writes, the first step must be a good specification of the model, the choice of the statistic method depends on this specification and on the available data. To ease the specification, I recommended a stepwise view, how price changes of products might feed through to other products/services.

To take the first step: In general, oil prices go rather quickly into the calculation of the prices for gasoline and diesel:

Cost of oil input (Can be assumed as a1*oil price

+Refining cost (incl. profit mark-up of the refinery)

+Mineral oil tax (and other specific taxes like CO2-tax)

+Cost of transport to gas stations

+Mark-up of gas stations

=Fuel price before VAT (input cost for other products)

+Value-added tax

=Price for final consumption (relevant for first-round effect on CPI)

For the specification of an equation for estimation, this has of course to be simplified. Assuming that part of the lower components are constant and other dependent on values, one could e.g. formulate an equation like:

fuel price (b.VAT) = (a1*oil price+a2*specific taxes)*(1+a3)+a4 =

=b1*oil price+b2*spec.t.+a4 or in percentage changes approximately.

%fuel price=b1*%oil price+b2*(spec.t./oil price-1)+b3 (should be near 0). As the weights for fuels in CPI remain constant for several years (and under the assumption, that they are similar for the whole time period considered), one could estimate the first-round effect on CPI by

%CPI=c1*%oil price+c2*(spec.t./oil price-1)+c3.

A specification like this, shows you which functional form to use for estimation and to hold "track" with the estimation in the sense, that you know what signs the coefficients should have and that you can also make rough calcultieons about the likely "true" values of the estimates.

For this step, I recommend an estimation for one or a few countries with monthly data (without lag and with max lag of 1).

You could do that for all steps of price feed-through, and finally come to a specification like

%CPI=c11*%oil price+c12*%oil price+...*c21*(spec.t./oil price-1)+...c3.

To this equation, one can add a variable showing lagged percentage increases of wages and you need a %wage equation (dependent on %CPI) to complete the model.

One problem is, that most of the lags are likely short-term, which means the main effects take place within less than half a year, with semiannual data they are without lags. I do not know, if you use average or end data. For both (but more for averages), the ci's depend on when the changes of the explaining variables have taken place. Maybe, you could include a dummy variable for that. Even a more severe problem is, that you have no good time series for the specific taxes on fuels, which are rather high in Europe and can, therefore have a considerable influence on CPI. Leaving-out this variable means a missing-variable problem with a high efficiency-loss of the estimation (high significance intervals) and possibly also a bias, if countries try to compensate high fluctuations of the oil price by changing taxes in the opposite direction.

You can, of course, add a demand variable. Because it concerns the effect on CPI, I would take private consumption instead of GDP. One should begin with looking at the correlations with different lags and after that trying a simple OLS estimate. Because of the assumed simple lag structure, one need not and should not apply ECM, VAR and other more sophisticated methods. But, I think, that John can give you better advice what concerns methods after you present us your specification and estimates.

Good luck!

It is difficult to say what might be causing the issue without more information about your specific model and data. However, it is generally recommended to use information criteria such as the Akaike Information Criterion (AIC) or the Schwarz Information Criterion (SC) to select the appropriate lag length for a VECM model. If you are getting strange results with these criteria, you may want to consider using other methods for selecting the lag length, such as cross-validation.

The AIC (Akaike Information Criterion) and SC (Schwarz Information Criterion) are used to select the optimal lag length for a VECM model. The AIC and SC are both measures of the goodness of fit of a model. The AIC is based on the likelihood function of the model, while the SC is based on the Bayesian Information Criterion (BIC). The optimal lag length is the one that minimizes the AIC or SC. If you are getting weird results for lag selection, you may want to try using other information criteria such as Hannan-Quinn Information Criterion (HQIC) or Final Prediction Error (FPE).