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4EK631 Econometric Modelling (2025/2026 Summer)

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Suppose the two variables, x and y, are integrated of order one, and their linear combination is tested using the Augmented Dickey–Fuller (ADF) test, yielding the following results: test statistic = −4.15 and critical value = −3.97.

You want to construct either

(a) an Error Correction Model (ECM), or

(b) a model in differences:

Which model is most appropriate?

Since both time series are integrated of the same order, we should use the ECM model right away, no matter what the ADF test tells us.

❌

We should use the model on differences since no cointegration relationship was found.

❌

Both time series are integrated of the first order. We should test the time series for cointegration, and the Breusch–Godfrey test is appropriate for this.

❌

Since both time series are integrated of the same order and there is a cointegration relationship, the ECM model is preferred.

✅

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What is the order of integration of the time series gdp from the Wooldridge package? Use a 5% significance level and the appropriate variant of the Augmented Dickey-Fuller Unit Root Test in the ur.df function from the urca package.

library('wooldridge')

library('urca')

data(inven)

inven$gdp

Since the different types of tests give different results, the test is inconclusive.

❌

The time series is integrated of the second or higher order.

❌

The time series is integrated of the first order, and the test needs to be executed twice to determine this.

✅

The time series is integrated of order zero, as the test statistic does not lie within the critical region.

❌

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Consider the real money (lrm1

) and real income

(

lny) time series from the urca

package. Both time series are integrated of

order 1.

Use a classical linear regression model and the Augmented Dickey-Fuller Unit Root (ADF) test via the ur.df

function to determine whether there is a

cointegration
relationship
between these two time series.

library(urca)

data(finland)

lm(lny ~ lrm1, data = finland)

Yes, a cointegration relationship is found

when a 1% significance level is used for the ADF test.

✅

No cointegration relationship is found even

when a 10% significance level is used.

❌

Yes, a

cointegration relationship is found when a 10% significance level is used for

the ADF test. However, no cointegration relationship is found using the 5%

level.

❌

Yes, a

cointegration relationship is found when a 5% significance level is used for

the ADF test. However, no cointegration relationship is found using the 1%

level.

❌

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Model the daily closing prices of DAX (Deutsche Aktienindex)

on the prices of FTSE (Financial Times Stock Exchange) using a

finite
distributed lag
model:

data(EuStockMarkets)

m3 <- lm(DAX[3:100] ~ FTSE[3:100] + FTSE[2:99] +

FTSE[1:98], data=EuStockMarkets)

Report the impact multiplier

(propensity)

using at least 2 decimal places.

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Suppose you want to model the stopping distances of cars (dist) based on their speed (speed). Your task is to test for the presence of outliers in the data using Grubbs' test.

library(outliers)

data("cars")

head(cars)

Use the grubbs.test

function from

the

outliers package and select the correct answer.

The lowest value of speed (4) and the highest value of dist

(120)

are significant outliers.

❌

There are no significant outliers

present in either variable

at a

1% significance level.

✅

There is no strict rule to decide, but some literature suggests that points for which Grubbs’s value is higher than 1 should be considered as outliers. Other literature provides a threshold of 4/n or 4/(n−k−1).

❌

At a

5% significance

level

, the variable speed includes at least one outlier, while dist does not.

❌

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Select all correct statements about Grubbs’ test. Note that multiple answers may be correct, and incorrect answers will result in point deductions.

Grubbs' test is based on the assumption of normality.

✅

For Grubbs' test, all variables need to be log-transformed.

❌

Grubbs' test is not a statistical test. T

here is no strict rule to decide, but some literature

suggests that points for which Grubbs’s value is higher than 1 should be

considered as outliers. Other literature provides a threshold of 4/n or

4/(n−k−1).

❌

Since the Grubbs' test detects influential observations in the model, the model must be estimated first.

❌

Grubbs' test detects one outlier at a time.

✅

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Your task is to model the stopping distances of cars (dist) using a classical linear model (lm function), with the speed of the cars (speed) as the explanatory variable.

data("cars")

head(cars)

Report the Cook's distance value for the most influential observation using the cooks.distance function.

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Select all correct statements about the finite distributed

lag model. Note that multiple answers may be correct. (A wrong answer will

result in negative points.)

The impact multiplier (propensity) is equal to the intercept.

❌

The standard error of the long-run multiplier in the finite

distributed lag model is calculated as the sum of the standard errors of the

individual effects in the model.

❌

The finite distributed lag model is designed to capture the

relationship between the dependent variable and lagged values of explanatory

variables.

✅

The impact multiplier is typically less accurately estimated

due to multicollinearity issues than the long-run multiplier.

✅

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Model the daily closing prices of DAX (Deutsche Aktienindex) on the prices of FTSE (Financial Times Stock Exchange) using a finite distributed lag model:

data(EuStockMarkets)

m3 <- lm(DAX[3:100] ~ FTSE[3:100] + FTSE[2:99] + FTSE[1:98], data=EuStockMarkets)

Report the long-run multiplier (propensity) using at least 2 decimal places.

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Your task

is to model the

linear trend in

the

flow of the River Nile

. Use

the following snippet of the script:

data("Nile")