Group Assignment I This document contains further instructions regarding your homework. It assumes you have read the original assignment. Your homework comprises two parts: 1. Decomposing GDP: you should decompose GDP according to the income and expenditure approaches. That means simply dividing the variable you are interested in by GDP. 1 For example, divide consumption by GDP to obtain the contribution of consumption to the national product. 2 You should then plot the evolution of these shares over time and analyse it (for instance, has consumption increased as a proportion of GDP, or remained constant, or varied?). You can make one graph per decomposition method, where you plot all the different components, as long as it is legible. In this part of the homework, you can focus on the two last decades. 3 You can also use annual data, instead of quarterly. 2. Analysing time series: you should analyse the evolution over time of the following variables: GDP, consumption, investment, employment, the unemployment rate and the price level. 4 Do not forget to deflate GDP, consumption and investment. 5 As concerns GDP, consumption, investment and employment, you should prepare four graphs per variable: the original series, its log, the trend of the log series and the cycle component. (See information about detrending below.) As concerns the unemployment rate and the price level, you need only plot the original series. You can combine different series on common graphs, as long as they are legible. Write a short comment on your findings (can you identify structural breaks? can you recognise important recessions or booms? are booms and recessions symmetric? etc.). You should use quarterly data (except for employment) 6 from at least as far back as 1980. You have to choose a detrending method for part 2. You have three options, detailed below. Whatever your choice is, justify it. After you have calculated the trend, subtract it from the original series to obtain the business cycle. 1. Linear regression: you fit a straight line, represented by the equation = + +. The first two terms, +, capture the trend, whereas stands for the cyclical component. Linear regressions ignore structural breaks, so they constitute only a very rough approximation and may produce weird results, such as years-long recessions. To compute the coefficients and on Excel, follow these steps: 1 Obviously, you need not deflate the series and you should not take logs. 2 Notice that the different components will not sum to a hundred, because you are not asked to look at exports and imports. 3 Or even on the past ten years, if you cannot find data for the income approach previous to the year 2000. 4 As measured by either the GDP deflator or the inflation rate, as you wish. 5 To deflate a series, multiply it by a hundred and divide it by the deflator. 6 You might obtain quarterly data for employment from the IMF (check the International Financial Statistics database, available from the library s website) or national statistics offices. (Extra points if you do.)
Take the log of real GDP (i.e. deflated GDP) and create an auxiliary time variable. Select two adjacent cells, side by side.
Enter the linear regression function (with the two cells selected!). The exact function depends on the language of your MS Office package. Mine is in Portuguese, so it is PROJ.LIN. In French, it is DROITEREG. In English, it is LINEST. It takes two mandatory arguments: the first is the dependent variable (GDP), the second is the explanatory variable (the time trend). Press CTRL+SHIFT+ENTER to return the results. The first cell contains the slope coefficient, whereas the second cell contains the constant. To compute the trend, multiply the time trend by and add.
Voilà. 2. Moving averages: you take the average of a certain number of quarters before and after the point in time you are interested in. This technique produces better trend estimates than linear regressions. However, there are two drawbacks: a) you lose observations; and b) you must select the number of quarters arbitrarily. The more quarters you include, the smoother your trend will be. Usually, a total of seven to eleven quarters (i.e. three to five quarters before the point you are considering and as many afterwards) is a good choice. Sum the quarters you will use to compute the average and divide by their number.
And that is it! 3. The Hodrick-Prescott filter: the HP-filter calculates a trend series (, ) so as to solve the following problem: min ( ) + ( ) ( ). The first term, ( ), captures the distance between the trend and the series. You do not want your trend to deviate persistently from the series, otherwise you might overestimate cycles and miss changes in the underlying growth trend. The second term, ( ) ( ), captures the volatility of the trend. Because it is a trend, it should be roughly constant and not move a lot from year to year. There is a trade-off between these two objectives. How much importance you assign to each of them is reflected by your choice of the weighting parameter. For quarterly data, you should set =1600.
Excel does not include a HP-filter utility. However, Kurt Annen developed an add-on to that effect. We have uploaded it on the ENTG and on the course s website (the file is called HPFilter.xla). Hence, as a first step, you ought to download it and add it to Excel.
Select the cells where the trend must be displayed. You must select as many cells as there are observations. Use the function HP to compute a HP-filtered trend. It takes two arguments: the series you wish to transform and the parameter. Press CTRL+SHIFT+ENTER to return the results.
Voilà.