The exercise deals with new car registrations in Denmark. Buying a car is in Denmark something that normally takes place in the spring or for many persons in the business sector just before Christmas with the car delivered March next year.

Problem 1

The folder problem 1 reports quarterly seasonal data from 1992Q1 to 2018Q1 on new car registrations, imports, the rate of interest and a trend variable. Data are NOT seasonally adjusted.

In addition, you will in the folder find the logarithmic transformation of the variables and a trend variable called TREND.

Buying a car normally involves loans, so if the rate of interest is low then it is cheap obtain finance for a car. It is more doubtful with the imports. Although all new cars are imported about half of all registrations are re-registrations of secondhand cars that not necessary affect the imports.

A. Provide time series diagrams for the variables NCARREG, IMPORTS and RATE, and a matrix of correlation. What is observed?

B. Estimate a regression model of the following form for the period 1992Q1 to 2018Q1:

NCARREG_t = J₀ +J₁IMPORTS_t + J₂RATE_t + J₃TREND + σ_t

Perform a model control for this regression.

C. To improve the model dynamics are introduced. The two variables to explain the car registrations are lagged by 4 period's equivalent to a year. The model to be estimated for the period 1993Q1 to 2018Q1 is now of the form:

NCARREG_t = J₀ +J₁IMPORTS_t-4 + J₂RATE_t-4 + J₃TREND + σ_t

Conduct a model control and investigate for the presence of autocorrelation by use of the Durbin Watson test. Save the residuals from this regression and call this variable RESID. Comment on the sign of the included variables. Assume a 10 percent level of significance. Compare also the model found under question B.

D. Expand the initial model above to an autoregressive (AR) model of order 4. This is an AR-4 model for the period 1994Q1 to 2018Q1. Estimate a model of the form:

NCARREG_t = J₀ + J₁IMPORTS_t-4 + J₂RATE_t-4 + J₃TREND + J₄RESID_t-1 + J₅RESID_t-2 + J₆RESID_t-3 + J₇RESID_{t -4} + σ_t

Why is it interesting for this data set to consider a model of order 4 relative to quarterly statistics?

Estimate the model and conduct a model control. Perform the Durbin Watson test. How are the plots of residuals? Compare with the findings under question C.

Conduct an LM-test where the model under H0 is the model estimated under question C, and the model under H1 is the model estimated under question D. Which model is the best? What happened to the autocorrelation?

E. Use the logarithmic transformed variables LNCARREG, LIMPORTS and LRATE to estimate an Error Correction Model (ECM). Explain the foundations for the ECM- model. Estimate first a long run regression model of the form:

LNCARREG_t = J₀ +J₁IMPORTS_t + J₂LRATE_t + σ_t

Save the residuals from this model and call them ECMRESID. Estimate second the short run ECM-model:

O₄LNCARREG_t = J₀ +J₁O₄LIMPORTS_t + J₂O₄LRATE_t + 6(LNCARREG_t-4 - Z₀ - Z₁LIMPORTS_t-4 - Z₂LRATE_t-4 ) + σ_t

Or

O₄LNCARREG_t = J₀ +J₁O₄LIMPORTS_t + J₂O₄LRATE_t + J₃ECMRESID_t-4 + σ_t

Here ? is the fourth order difference operator. Is the ECM-term β₃ negatively significant as should be the case for the model to be valid? How good is the model, and how is the model control?

Problem 2

The folder problem 2A reports quarterly NOT seasonal adjusted data from 1992Q1 to 2018Q1 on the new car registrations in Denmark. In addition, 3 seasonal dummies can be found and the logarithmic transformation of the new car registrations in Denmark

A. Examine for quarterly deterministic seasonality and estimate a model of the form

NCARREGt = J₀ +J₁D2_t + J₂D₃t + J₃D₄t + σ_t

How good is the model? Is deterministic seasonality observed? Is it true that the spring is the high season for new car registrations?

B. Examine for stochastic quarterly seasonality by use of the HEGY-test

Set up the hypotheses and find the transformed variables. Plot the transformed variables and comment. What is the outcome of the test?

The folder problem 2B reports monthly NOT seasonal adjusted data from 1992M1 to 2018M3 of the new car registrations in Denmark. In addition, 11 seasonal dummies can be found.

C. Examine for deterministic monthly seasonality and estimate a model of the form

MNCARREG_t = J₀ +J₁D2_t + J₂D₃t + J₃D₄t + ... + J₁₁D₁₁t + J₁₂D₁₂t + σ_t

(I.e. all 11 seasonal dummies included). Is it true that March, May and June are special? Compare also with the model estimated on the quarterly data in question A. Is any information lost by using only the model estimated under question A?

Problem 3

Maybe you have been driving in a car, and you have experienced that not only the other car drivers, but also the other truck drivers on the road are occupied with things that not are related to focused driving for example talking in hand held phone or even worse sending SMS's. In addition, some drivers may also be eating, having the attention to tune in the radio etc.

The folder problem 3 reports statistics from 16 European countries on the shares of truck drivers that are performing distractive activities while driving. Statistics has been collected by traffic police in a number of European countries during February 2018. Here the police have given increased attention to these events causing an increased probability of serious traffic accidents.

Data are divided in to three groups according to traffic in the city areas, land areas and motorways. Data reports the share of drivers with distractive performance out of all drivers. In addition the data are divided into four segments of four countries each namely Scandinavia, small Western countries, large Western countries and East European countries. Data are given in percent shares.

A. Examine the hypothesis that distractive driving is more likely to occur on motorways. Use the simple ANOVA-method. Set up the hypotheses for the test, calculate the ANOVA-table, and provide an interpretation of the outcome. Provide a supplementary analysis.
B. Each group of countries can be considered as a segment. Set up and conduct a two- way ANOVA analysis for the data set. Describe the method and set up the hypotheses behind the test. What is the outcome? Do we observe interaction or segmentation among the two factors? What is the interpretation? Which group(s)/segments of countries are of interest with regard to both factors? Can a ranking be undertaken?

Problem 4

Let us continue our trip on the roads with a break and a cup of coffee! The coffee is provided by a coffee machine that from time to time needs service. When the indicator on the machine lights up then it is time for a service. The producer guarantees that the coffee machine can provide minimum 13,000 cups of coffee before each service. However, sometimes this occurs long time before the 13,000 cups of coffee has been provided. However, the producer guarantees that this will be the case in less than 1 percent of the cases.

The folder problem 4 reports statistics from 25 coffee machines. The table gives the number of coffee cups made before the service indicator showed that a service was needed.

A. Examine the symmetry of the dataset and conduct the Bowman-Shenton test. Set up the hypotheses, describe and perform the test. Is the material Normal Distributed?

B. Set up a sign-test and examine a hypothesis stating that the median is higher than 13,000. Set up the hypotheses, describe and perform the test. What is the outcome? What about the guarantee and the 1 percent?

Problem 5

The folder problem 5 contains a data set of 400 randomly selected incomes from Denmark.

A. Conduct a descriptive statistical analysis for the variable DKINCOME. The analysis should contain a histogram and commenting on the symmetry. Describe the shape of the distribution of data

B. Compile by use of simple random selection a sample of 10 percent or 40 observations of the variable DKINCOME

C. Compile by use of stratified random sample with systematic sampling inside each stratum a sample of 10 percent or 40 observations of the variable DKINCOME. Use the histogram drawn under question A as a point of departure for setting up the strata's

D. Provide a comparison of your results found under question A-C. Comment of your findings. Use descriptive statistics and a discussion of the shape of the samples compared to the distribution of the total data set. Discuss the implications of the presence of eventual outliers on the process of sampling.