Stationarity And Unit Root Testing Finance Essay

Determining the type of stationarity of the series can be done by sing the clip tendency arising in the variables. The first measure is to run a arrested development on the different log series with a changeless i?? and a tendency i?? plus an error term, i??t

yt = i?? + i??t + i??t

log ( OIL ) = + 2.443 + 0.02265*Trend

( SE ) ( 0.0726 ) ( 0.00139 )

The contrast between trend- and difference-stationary could be seen by ocular review, and make up one’s minding whether the the detrended A· has yielded a stationary procedure

The diagram displays unmistakably non-stationary remainders, proposing that we should handle log ( OIL ) as difference-stationary. Equivalent consequences were so found for the staying 5 energy markets, GASOIL, JK, NAPH, LSFO and HSFO.

Unit Root testing, finding the order of integrating

For an analyst gauging clip series theoretical accounts, the features of stationary procedures are extremely desirable. By executing a unit root tests we can uncover two utile characteristics

The series ‘ order of integrating I ( vitamin D ) , how many times the theoretical account requires to be first-differenced to accomplish stationarity.

Given difference-stationary series, we want to find if they have stochastic or deterministic tendencies.

By following the guidelines defined in Holden and Perman 1994, we are able to reply these inquiries. By ocular review from the old graph, it is clear that yt is a turning procedure. A changeless and tendency are therefore included in the first arrested development. The process for log ( OIL ) is outlined below

Yt = a + bt + pYt-1 + et

By presenting the slowdown operator L ( LYt = Yt-1 ) and rewriting this to

( 1-pL ) Yt = a + bt + et |p| & lt ; 1

We can see that the multinomial ( 1-pL ) has a individual root equal to 1/p which equals one if p = 1 and is when the procedure is non-stationary. This is the Dickey Fuller ( DF ) void hypothesis.

H0: P – 1 = 0 vs. Ha: |p| & lt ; 1

However, when running ( 1.1 ) I find the mistake footings holding some issues related to consecutive correlativity ( AR 1-5 trial ) . The redress is to present the Augmented Dickey Fuller ( ADF ) trial which can include lagged dependent variables. Step 1 is therefore to happen an appropriate procedure such that remainders are white noise, avoiding autocorrelation. By misspecification trials I find that four lagged reliable variables are suited for quarterly informations ( see AR 1-5 trial below ) . Taking the first difference of Yt in ( 1.1 ) yields the arrested development to be used to prove for a unit root.

I”Yt = a + bt + pYt-1 + i?±1I”Yt-1 + i?±2I”Yt-2 + i?±iˆ?I”Yt-3 + i?±iˆ?I”Yt-4 + et

PcGive end product by patterning the above equation by OLS appraisal method






log ( OIL ) _1






Dlog ( OIL ) _1






Dlog ( OIL ) _2






Dlog ( OIL ) _3






Dlog ( OIL ) _4


















AR 1-5 trial: F ( 5,73 ) = 1.0554 [ 0.3922 ]

The following stairss are proving the relevant hypotheses by an F-Test and utilizing the i?¦iˆ provided by Dickey and Fuller, 1981. I have run the Exclusion limitations specification trial to achieve the trial values


Test/Restrictions on

End product

Critical Value at 5 percentage. N = 100



Tendency and Yt-1

H0: i??iˆ¬i??iˆ¬i?? = a,0,0 vs. H1: i??iˆ¬i??iˆ¬i?? =/ a,0,0

Subset F ( 2,78 ) = 3.7217 [ 0.0286 ] *

6.49 ( i?†iˆ?iˆ©

Do non reject joint H0 of unit root and no clip tendency


Changeless and Yt-1,

H0: i??iˆ¬i??iˆ = 0,0 vs H1: i??iˆ¬i??iˆ =/ 0,0

Subset F ( 2,82 ) = 0.81270 [ 0.4473 ]

4.71 ( i?†iˆ±iˆ©

Do non reject joint H0 of unit root and no impetus

I”Yt = pYt-1 + i?±1I”Yt-1 + i?±2I”Yt-2 + i?±1I”Yt-3 + i?±1I”Yt-4 + et

Based on the three formal trials, we have established I”Yt to be a stationary procedure, I ( 0 ) . Subsequently, we can reason that the series of log ( OIL ) is I ( 1 ) and a random walk.

Performing matching trials for the other variables outputs unsurprisingly similar consequences as good, displayed below.

Figure: Trial for unit root at the log degree

Figure: Trial for unit root at the dlog degree

The decisions from these trials are that all the energy monetary value series are unity, and therefore can be differenced one time to acquire stationarity. Perceptibly, the rejection of any deterministic footings is a spot surprising by the old informal analysis. However, to verify that the series are I ( 1 ) we could farther difference, Internet Explorer. I”I”yt. The result of this confirms that the series are stationary merely after taking the first difference one time. Furthermore, from an economic point of position, a daze contagious disease will hold relentless effects on series with unit root, such as our instance in the energy market. An illustration may be seen around the period of the new millenary, with the latent Asiatic Financial crisis every bit good as geopolitical convulsion in the Middle-East.

Question 2 – Trial for cointegration to separate between a cointegrated relationship and a specious arrested development instance.

For two series to be cointegrated, we say that the long-run belongingss must be comparable. If a additive combination of ( any ) two series are yt ~ ( vitamin D ) and xt ( vitamin D ) so if there exists a vector beta such that the error term is of a lower order of integrating, I ( d-b ) , so yt and crosstalk is said to be cointegrated of order ( vitamin D, B ) . If the error term of the arrested development is stationary, so the variables in inquiry are said to be cointegrated. In other words, there exists a long-term relationship towards which they ever will return to. This besides entails that cointegrated theoretical accounts may be utile in calculating future motions or monetary values.

From the old inquiries we established that both series are I ( 1 ) , fulfilling the first standards. The following measure is to find if there exists a long tally equilibrium relationship by a OLS calculator

Yt = b0 + b1Xt + et

Outputs the undermentioned calculators

log ( OIL ) = – 0.2957 + 1.029*log ( GASOIL )

( SE ) ( 0.0381 ) ( 0.0102 )

Then we save the remainders from this arrested development to run a ADF trial for unit root, CoInt_Res: ADF trials: t-adf: [ -4.544 ] ** ( T=84, Constant ; 5 % =-2.90 1 % =-3.51 )

We reject the nothing of a unit root, and conclude that there exists a additive combination of the series with a lower integrating order. Consequently, this theoretical account has an ECM preparation such that it is immune from the specious arrested development job.

ECM = log ( OIL ) + 0.295689 – 1.02878*log ( GASOIL ) ;

When we say that two clip series spuriously correlative, a apparently related relationship will be inferred by a arrested development of the two, where one should non. This can be due to high autocorrelation which can do the analyst accept a important incline coefficient when in fact the two variables are unrelated. Obviously, this inaccuracy can take to serious illation mistakes, and the two documents of Ferson et Al ( 2003 ) discusses this issue related to fiscal economic sciences every bit good as stock returns.

Ferson et Al ( 2003a, B ) besides suggests caution about a 2nd issue, informations excavation ; the acknowledgment of forms in big graduated table informations sets. The concern is when informations excavation reinforce the job of specious arrested development, by picking instruments that show superior continuity. The remainder of the paper is spent showing a theoretical account which applied to nine of the major surveies on foretelling stock returns, lineations marks of the above issues.

A specious arrested development will give an wrong t-value, bespeaking a important relationship between independent variables. Through simulation the writers ‘ discoveries that it is the denominator of the t-value that is the cause of mistake. The standard mistakes are non able to right account for the consecutive dependance and hence biased. Given inconsistent standard mistakes, the experiment finds that 7 of the 17 t-statistics are falsely said to be important. Consequently, the venereal disease. mistakes should be corrected, a “ stochastic detrending ” of the lagged variable is suggested, to do standard mistakes efficient and indifferent. Similarly, when including informations excavation for high R^2 forecasters, the narrative gets worse.

Interpreting the consequences from the trials and the documents in an economic position, cointegrated series should travel closely together, and the divergency between them will prevail stationary over clip. If, likewise to stock returns, energy trade goods ‘ expected returns remain stable over clip, they should be unpredictable ( rho = 0 ) . However, as the survey of Ferson et Al ( 2003 ) suggests, taking an appropriate redress to evade specious arrested development, every bit good as a cointegrated relationships, could give non-trivial prognostic power for the theoretical account.

Question 3 Multivariate Time Series

The impression of cointegrated plus monetary values as found above is at odds with the efficient market theory ( EMH ) . While the EMH states that since all information available is reflected in the monetary values, consistent anticipations or merely crushing the market ( extra returns ) , is non possible. Perversely to this, the being of cointegration and ECM suggests a prognostic power on returns, from the current, short-term divergence to its long-term relationship. Richards ( 1995 ) tests this hypothesis of cointegration across several major stock market indices and whether there exists a common stochastic tendency between them.

With stock returns virtually ever are found to be I ( 1 ) and being comparatively easy to regress, empirical surveies on cointegration between different companies or stock indices has been comparatively popular. One of the most noticeable surveies is the 1 of Kasa ( 1992 ) , where four cointegrating vectors were found between national stock monetary value indices. However, these consequences are disputed given the unusual inclusion of 10 slowdown periods every bit good as a little sample period, and every bit demonstrated subsequently, doing over-parameterization.

Johansen attack

The chosen slowdown construction of our theoretical account has major importance ; the more slowdowns included, the easier it is to reject the nothing of no cointegration. We besides pay attending to happening the appropriate figure of slowdowns to utilize in the multiple-equation dynamic theoretical account to decide any issues related to consecutive correlativity. The determination on how many slowdowns to utilize will be based on the Schwarz standard ( SIC ) . With quarterly informations an applicable scope of slowdown is between 4-6 slowdowns. By the undermentioned consequences, we choose the theoretical account by the argmin Schwarz. With

[ insert equation to utilize for johansen attack, sum-sign for slowdowns, p.77 Harris, notes? ]









Vector AR 1-5 trial: F ( 180,120 ) = 0.94842 [ 0.6289 ]

No consecutive correlativity.

N = 6

Finding the cointegration relationships in a multivariate instance ( VAR ) is done by happening the ‘cointegration ‘ rank of the matrix PI and so measure if any of the characteristic root of a square matrixs of PI are zero. The void hypothesis used is

H0: R & lt ; = r0 against H1: R & gt ; r0, utilizing the hint statistic. The ( merely interesting ) instance where PI is I ( 0 ) is when up to ( n-1 ) cointegration dealingss exists. Furthermore, this tells us there will be r additive independent cointegrating vectors and therefore ( n-r ) “ common stochastic tendencies ” driving the system. Runing this trial yields the undermentioned hint consequences

I ( 1 ) cointegration analysis, 1991 ( 1 ) – 2012 ( 2 )

H0: rank & lt ; = Trace trial [ Prob ]

0 120.79 [ 0.000 ] **

1 75.633 [ 0.015 ] *

40.052 [ 0.223 ]

We start by proving the void hypothesis of R = 0, which is rejected at the 5 % degree. We so continue to r=1 and so forth. We stop when we can accept the nothing of cointegration, which here is at R = 2. As a consequence, we conclude that the figure of cointegrating vectors ( 2 ) gives us 4 common stochastic tendencies when we have 6 variables in consideration.

By similar process, Richards simulate the Kasa ( 1992 ) survey to look into its sensitiveness to dawdle included in the theoretical account. Based on this Richards finds merely specious rejections of the void hypothesis of no cointegration, more in alliance to the EMH. Although few, there were still peculiar instances demoing less than perfect correlativities, on both company particular and national degree, an indicant of predictability. The paper besides includes a survey of a “ winner-loser scheme ” , which outcome suggests a mean-reverting tendency for assets from over- to under-performance and frailty versa. The possibility of cointegration between oil and gasoil monetary values should non be disregarded by the findings in Richards ( 1995 ) paper. Though our series apparently have similar characteristics to the variables in Richards survey, they are nevertheless trade goods, non stock indices nor managed companies. The survey is hence non accurately movable.

B ) Contagious disease.

Reviewing the results from major macroeconomic dazes throughout the last decennaries, one can comprehend correlative motions between different stock indices, proposing contagious disease exists. This phenomenon is defined by Forbes and Rigobon ( 2002 ) as “ a important addition in cross-market linkages after a daze to one state ( or group of states ) ” , with weight laid on addition. By this categorization, if there exists high cross-market correlativity in all provinces of nature ( pre and station daze ) , the state of affairs is termed mutuality. The paper examines old findings of contagious disease on a national degree after four important macroeconomic events occured.

There are several methodological analysiss to mensurate how macroeconomic dazes are spread globally. The most straightforward trial is the cross-market correlativity coefficient trial. This is the trial I will utilize to prove for contagious disease with oil to the other markets. The trial process is outline below

Yt = a + bXt + et with iid et

Second, we divide the full clip period into two subsamples, a stable and convulsion, where Var ( Xturmoil ) & gt ; Var ( Xstable ) . Correspondingly, the cross-market covariance will besides be higher in the latter group. Our stable period is set from 1990 ( 1 ) up to the 2nd one-fourth of 1997. Our period of convulsion is initiated by the Asiatic Financial Crisis and geopolitical hurts around Iraq get downing April 1997, and besides includes the dot com bubble and the more recent planetary meltdown, up to 2012 ( 2 ) .

In order to execute a t-test on whether correlativity during the convulsion is significantly higher than the full sample, the rho and sigma has been calculated for each period. The undermentioned t-test has been run to get at the concluding column.

As can be seen from the last column, we can non reject the nothing of no contagious disease between oil markets and the five other energy markets. Compared to consequences in the paper, these consequences are non surprising, with the chosen slowdown period and sample period. However, if contagious disease was found, as in the instance of Kasa ( 2002 ) , the inquiry of conditioned discrepancy and rectification for this would be the following measure in the trial process. If the exogenic variable Crosstalk has heteroskedastic discrepancy, Yts ‘ discrepancy is said to be conditioned, which can do deceptive illations. To set for this prejudice, Forbes and Rigobon ( 2002 ) pull strings the correlativity coefficient to go unconditioned of this. A caution here is that this assumes we have non omitted any variables or endogeneity. From this, Forbes and Rigobons ‘ finding of fact is that predating trials are non blunt, alas biased and inaccurate, due to heteroskedasticity in market returns. In their adjusted simulation there is merely one instance where contagious disease is found under unconditioned cross-market correlativity. However, the survey suggests markets move jointly during all provinces of nature, a phenomenon labelled mutuality. The below chart of the different energy markets suggests a similar decision.

Question 4

Trial for ARCH effects by Lagrange multiplier trial

A trial the presence of ARCH effects is by the Lagrange multiplier trial, introduced by Robert F. Engle in 1982. With fiscal clip series there could be ground to believe that the discrepancy has ‘memory ‘ , or in other words be a map of former mistake footings. By a ocular review of the monthly log returns, there is an indicant of periods with little volatility and other with higher volatility. This is a clear intimation of ARCH consequence nowadays in the energy markets. On the lower left of the below panel, comparing the Crude oil distribution to a normal exhibits a narrower and right skewed extremum. Fatter dress suits are besides apparent. Finally on the right manus side is the autocorrelogram of both the log and squared log returns. These show mark of consecutive correlativity in both minutes.

The first measure is to place the AR procedure to obtain the squared returns from. This is chosen under argmin for Schwarz standard ; a AR ( 1 ) theoretical account yielded the lowest value.

Uhat^2t = a0 + a1ut-1 +et

Under the void hypothesis we have absence of ARCH effects: a1 = 0. The trial statistic to be used is the figure of observations ( T ) times the R^2. The consequences are outlined in the tabular array below


R^2 ( # of observations )


Chi^2 C.V. = 5.991



0.130965 ( 268 )

35.09862 **

Rejection of no ARCH effects present


0.0598597 ( 268 )

16.0423996 **

Rejection of no ARCH effects present


0.113247 ( 268 )

30.350196 **

Rejection of no ARCH effects present


0.0693118 ( 268 )

18.5755624 **

Rejection of no ARCH effects present


0.0559427 ( 268 )

14.9926436 **

Rejection of no ARCH effects present

We conclude from both analyses that all the discrepancy in energy markets contain some ARCH effects, bespeaking that farther GARCH modeling is so appropriate.

Testing for GARCH effects.



Schwarz standard











GARCH ( 1,1 )







GARCH ( 1,1 )







EGARCH ( 1,1 )







IGARCH ( 1,1 )







RiskMetrics ( fifty = 0.94 )








The highlighted values are the minimal SIC values of the five theoretical accounts. After the appraisal is complete we examine the GARCH theoretical accounts selected, which are the changeless conditional- mean and discrepancy theoretical account that best fits the clip series.

Estimated Parameters Vector for d_LSFO:

[ 0.010429 ; 0.006384 ; 0.364279 ; 0.079023 ]

yt = 0.0010429 + et

sigma^2t = 0.006384 + 0.364279sigma^2t_1 + 0.79023et^2t_1

Finally, we compare the the old Engle ARCH trial consequences with the GARCH theoretical account,

ARCH 1-2 trial: F ( 2,262 ) = 0.30240 [ 0.7393 ]

ARCH 1-5 trial: F ( 5,256 ) = 0.55467 [ 0.7347 ]

ARCH 1-10 trial: F ( 10,246 ) = 0.38353 [ 0.9531 ]

The comparative high chance value to accept no ARCH effects indicates that the theoretical account is satisfactory in capturing the GARCH effects in the energy markets.

July 23, 2017