Question

(a) Discuss the procedure in carrying out the Johannes Cointegration.

(b) Discuss how one can undertake the Eagle Granger error regression model.

(c) In empirical research heteroskedacity is not a nusance because it offers a research to re-estimate the model. Discuss

Answer 1

a.Cointegration is a statistical property possessed by some time series data that is defined by the concepts of stationarity and the order of integration of the series. A stationary series is one with a mean value which will not vary with the sampling period. For instance, the mean of a subset of a series does not differ significantly from the mean of any other subset of the same series .The significance of cointegration analysis is its intuitive appeal for dealing with difficulties that arise when using non-stationary series, particularly those that are assumed to have a long-run equilibrium relationship. For instance, when non-stationary series are used in regression analysis, one as a dependent variable and the other as an independent variable, statistical inference becomes problematic.

Johansen Test:

Since we want to form a stationary price series out of an arbitrary number of price series’ we can rewrite equation 1 using vectors and matrices for the data and there coefficients: ΔY(t)=ΛY(t−1)+M+A1ΔY(t−1)+...+AkΔY(t−k)+εt (2) Thus, Λ and A are matrices and Y(t) is in vector representation. Similar to the argumentation in subsection 2.1 (see also (Chan, 2013)) we need to test whether our Nullhypothesis that Λ = 0 (hence, no dependence between previously obtained and recent data and therefore no cointegration) can be rejected and at what certainty (Johansen, 1991). To actually find the linear coefficients for constructing a stationary price series a Eigenvalue decomposition ofΛ is applied. Let r be the rank ofΛ (r = rank(Λ)) and n the number of price series’ to be cointegrated than the Johansen test (Johansen, 1991) performs an Eigenvalue decomposition ofΛ and test whether we can reject the n nullhypothesis’ r≤0, r≤1,r≤2, ..., and r≤n−1. Thus, for an arbitrary positive integer n (the number of price series to be cointegrated) the Johansen test calculates the test statistics for the null-hypothesis’ that r ≤ i;wherei = 0,1,...,n−1. If all n−1 null hypothesis are rejected within a high degree of confidence (e.g., α = 0.05) we may concludethatindeedr =n.Therefore,theJohansentestborrowstheideafromthePCA (principle component analysis) and transforms our data using linear combination to a new coordinate system. Furthermore, since we basically applied PCA we now may use the eigenvector corresponding to the highest eigenvalue for our hedge ratios. Critical values for the test statistic results can be found in (Johansen, 1991).

b.The Journal of Econometrics and Econometrica are the two journals that contain the most cited papers in the econometrics discipline. These citation classics have in common that they mainly concern econometric techniques for the analysis of time series variables. By far the best cited time-series econometrics paper is Engle and Granger (1987). The Nobel-worthy concept of cointegration had already been introduced in Granger (1981), but the Econometrica paper in 1987 meant an explosive take-off for this novel idea. Many academics and practitioners resorted to the use of the cointegration technique, and theoretical developments in the area covered quite some space in econometrics conferences all over the world. A glance at the programs of the Econometric Society meetings in the 80s and 90s of the previous century, which can be found in back issues of Econometrica, shows that a large number of sessions were dedicated to just “Cointegration”. Even today there still are workshops and sessions during conferences where new developments in cointegration are being discussed.

Circumstances :

First, with the introduction of the influential book of Box and Jenkins (1970) there emerged an increased interest in analyzing time series data and using rather simple models for out-ofsample forecasting. Indeed, the proposed ARIMA models turned out to deliver high quality forecasts, and in fact, in those days these time series forecasts were observed to be much better than those from large-scale macro-econometric models containing hundreds of equations and equally sized numbers of unknown parameters. Even though it can be argued that large-scale simultaneous equations models can be written as VAR models, which in turn can be written as ARIMA type models (Sims, 1980 and Zellner and Palm, 1974), the sheer infinite distance between large models and ARIMA forecast schemes created a need for “models in between”. Possible candidates for this were the single-equation error correction models such as the well-known Davidson et al. (1978) consumption function. In fact, as discussed by Granger (2009), it was exactly the confrontation of such models with unit-root processes (as discussed next) that led to the notion of cointegration.

As we have stated, the regression of nonstationary series on other series may produce spurious regression. If each variable of the time series data is subjected to unit root analysis and it is found that all the variables are integrated of order one, I(1), then they contain a unit root. There is a possibility that the regression can still be meaningful (i.e. not spurious) provided that the variables cointegrate. In order to find out whether the variables cointegrate, the least squares regression equation is estimated and the residuals (the error term) of the regression equation are subjected to unit root analysis. If the residuals are stationary, that is I(0), it means that the variables under study cointegrate and have a long-term or equilibrium relationship. The Engle-Granger method is based on the idea described in this paragraph. In the two-step estimation procedure, Engle-Granger considered the problem of testing the null hypothesis of no cointegration between a set of variables by estimating the coefficient of a statistic relationship between economic variables using the OLS and applying well-known unit root tests to the residuals to test for stationarity. Rejecting the null hypothesis of a unit root is evidence in favour of cointegration.

There is vast literature that explores whether spot and future prices for oil are linked in a long run relationship. One particular study was undertaken by Maslyuk and Smyth (2009) to examine whether crude oil spot and future prices of the same and different grades cointegrate. The null hypothesis of no cointegration was tested against the alternative of cointegration in the presence of a regime shift on series monthly data from the United States Western Telematic Inc. and United Kingdom Brent using the two-step estimation procedure. The results revealed that there is a cointegration relationship between spot and future prices of crude oil of the same grade, as well as spot and future prices of different grades. Results further indicated that spot and future prices are governed by the same set of fundamentals, such as the exchange rate of the US dollar, macro economic variables, and the demand and supply conditions, which are similar and interrelated for crude oil in North American and European markets.

c.A key goal in empirical work is to estimate the structural, causal, or treatment effect of some variable on an outcome of interest, such as the impact of a labor market policy on outcomes like earnings or employment. Since many variables measuring policies or interventions are not exogenous, researchers often employ observational methods to estimate their effects. One important method is based on assuming that the variable of interest can be taken as exogenous after controlling for a sufficiently large set of other factors or covariates. A major problem that empirical researchers face when employing selection-on-observables methods to estimate structural e⁄ects is the availability of many potential covariates. This problem has become even more pronounced in recent years because of the widespread availability of large (or high-dimensional) new data sets.

We established asymptotic normality of the OLS estimator of a subset of coeffi cients in high-dimensional linear regression models with many nuisance covariates, and investigated the properties of several popular heteroskedasticity-robust standard error estimators in this high-dimensional context. We showed that none of the usual formulas deliver consistent standard errors when the number of covariates is not a vanishing proportion of the sample size. We also proposed a new standard error formula that is consistent under (conditional )heteroskedasticity and many covariates, which is fully automatic and does not assume special, restrictive structure on the regressors. Our results concern high-dimensional models where the number of covariates is at most a non-vanishing fraction of the sample size. A quite recent related literature concerns ultra high-dimensional models where the number of covariates is much larger than the sample size, but some form of (approximate) sparsity is imposed in the model; see, e.g., Belloni, Chernozhukov, and Hansen (2014), Farrell (2015), Belloni, Chernozhukov, Hansen, and Fernandez-Val (2017), and references therein. In that setting, inference is conducted after covariate selection, where the resulting number of selected covariates is at most a vanishing fraction of the sample size (usually much smaller). An implication of the results obtained in this paper is that the latter assumption cannot be dropped if post covariate selection inference is based on conventional standard errors. It would therefore be of interest to investigate whether the methods proposed herein can be applied also for inference post covariate selection in ultra-high-dimensional settings, which would allow for weaker forms of sparsity because more covariates could be selected for inference.