Posted: September 16th, 2015

Calculating the mean and variance.

 

In this Case Study, you will look at modelling the Australian dollar vis-á-vis the US dollar exchange rate, using a conventional monetary model. The data you will use is in the file called “The Monetary Model.xls”. This contains the following data: ‘S(AUD/USD)’ , which is the Australian dollar per US dollar exchange rate; ‘Aust. MS’ , which is the level of the Australian money supply; ‘US MS’ , which is the level of the US money supply; ‘Aust. GDP’ , which is a measure of Australian gross domestic product (GDP); ‘US GDP’ , which is a measure of US gross domestic product (GDP); ‘Aust. IR’ , which is a measure of short-term Australian interest rates, and ‘US IR’ , which is a measure of short-term US interest rates. Data on these variables were collected from the IMF’s International Financial Statistics database, and from the Reserve Bank of Australia’s website.

 

To begin the empirical analysis, make sure you take the natural logarithm of S(AUD/USD), Aust. MS, US MS, Aust. GDP, and US GDP. Leave the interest rate variables, Aust. IR and US IR, in their raw form. Also, use a level of significance of a = 0.05 for all tests in this study.

 

• Calculate the correlation between the spot exchange rate and all the other variables. Which variable has the strongest correlation with the exchange rate? I will call this variable “PREDICTOR” in the questions that follow.

 

• Calculate the mean, median and mode of the spot exchange rate. What is the difference between these types of “averages”? State what you think is the most appropriate average for this set of data, giving brief reasons for your answer.

 

• Draw a histogram of the spot exchange rate. Calculate the variance, measure of skewness and kurtosis. Compare the values of the median and mean. Do they indicate the data is symmetric, and is this confirmed by the value of the skewness? Comment on the value of kurtosis. Based on these results, say why you believe that the spot exchange rate does, or does not, have a normal distribution.

 

• Calculate the mean and variance of the PREDICTOR for the time period 1984:1 to 1992:4, and then for the period 1993:1 to 2013:4 separately. Using these figures, calculate the variance of the difference between the sample means for the period 1984:1 to 1992:4, and then for the period 1993:1 to 2013:4 (You can assume that the covariance is zero). In the first quarter of 1993, the Reserve Bank of Australia adopted an inflation targeting regime. Test to see whether the adoption of an inflation targeting regime has had any impact on PREDICTOR, that is, test to see whether the difference is significantly different from zero. Under the conditions discussed in part (e), the standardised value of the difference will have a Student’s-t distribution with n-2 degrees of freedom (Subtract 1 for each sample mean calculated).

 

• The test in part (d) is only reliable if both samples are large, or if a sample is small, it has a normal distribution. Do you think the test in part (d) is reliable? Give brief reasons for your answer.

 

 

In this Case Study, you will be looking at the effect of the 2010 Federal Election on the Australian stock market. In the file called “2013 Federal Election.xls”, there is daily data from the 1^st of January 2013 until the 31^st of December 2013, on the Australian All Ordinaries index and the MSCI world index. You will be looking at two important dates: the day the stock market opened after the election was called, that being the 31^st of January 2013, and the day the market opened after Tony Abbott’s election win, which was the 16^th of September, 2013.

• Obtain line graphs for both over the whole period and comment on any similarities or differences.

 

• Calculate the continuous returns on both indices. Obtain histograms and descriptive statistics up to, and including the 30^th of January, between the 31^st of January and the 13^th of September, and from the 16^th of September onwards. Comment on any similarities or differences between them.

 

• For the Australian All Ordinaries index, calculate the 95% confidence intervals for the mean return in the three periods given in part (b).

 

• Assume that the returns on the All Ordinaries index has a normal distribution. Using the means and standard deviations previously calculated for each of the three periods, find the probability of getting a negative return on a day selected at random.

 

• Write some brief notes, in point form, describing your results for Case Study Two. Be sure to report any conclusions that you have made, and submit these. You will use them in a later tutorial when you write your final report.

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CASE STUDY ONE – continued

 

In this section, you will estimate a simple linear regression model to fit the exchange rate. From Assignment One, choose the input variable that has the strongest correlation with the exchange rate that you believe will produce a good model (this is the variable that we called “PREDICTOR”).

 

Following on from Assignment One, use the same data set and a level of significance of a = 0.05 for all tests in this study.

 

• Draw the scatter diagram for these two variables.
• Estimate the model to predict the exchange rate. How much of the variation in the exchange rate is explained by your model? Is this statistically significant? Test to see whether the intercept is required in your model.

 

• Obtain a graph of the residuals vis-á-vis the independent variable for part (b). Is there any evidence of problems?
  CASE STUDY TWO – continued

In the next stage of this study, you will estimate betas for the Australian stock market vis-á-visthe world using the data in the file called, “2013 Federal Election.xls”. You will need the returns you calculated from a previous tutorial. You will also use the three sub-periods you used before, that is, the period before the election was called, the period during the election campaign, and the period after the election result. In this analysis, you will treat Australia as the share and the world as the market

(a) Draw scatter plots of the three pairs of returns you will use for estimating the market models.

(b) For the three sets of returns in each of the sub-periods find the betas using the Market model

 

Here the dependent variable is the ‘raw’ or ‘actual returns’ () for Australia and the independent variable is the ‘actual market returns’ (). Once you have estimated these betas use the ‘F’ statistic and the value of R² to evaluate these estimated models.

 

(c) For each of the sub-periods, test to see whether Australia is passive or neutral using a = 0.05. That is, test whether .

 

(d) For the first sub-period, estimate the Scholes-Williams Beta and Dimson’s Beta. Do you think you need to use this sort of adjustment here?

 

(e) Write some brief notes, in point form, describing your results for Case Study Two. Be sure to report any conclusions that you have made, and submit these. You will use them in a later tutorial when you write your final report.

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