REGIONAL WORKSHOP ON TRAFFIC FORECASTING AND ECONOMIC PLANNING

Transcription

1 International Civil Aviation Organization 27/8/10 WORKING PAPER REGIONAL WORKSHOP ON TRAFFIC FORECASTING AND ECONOMIC PLANNING Cairo 2 to 4 November 2010 Agenda Item 3 a): Forecasting Methodology (Presented by the Secretariat) Introduction 1. Under this agenda item, among other topics, model development using regression analyses will be discussed in detail. Hands-on exercises will be conducted with the participation of attendees with the objective of achieving a greater understanding of this technique and its application in the development of forecasts. Attached is a step-by-step approach to regression analysis using Microsoft Excel in the form of a user manual. (27 pages)

2 - 2 - Intentionally blank

3 - 3 - A STEP-BY-STEP APPROACH TO REGRESSION ANALYSIS USING MS EXCEL

4 - 4 - Outline 1. Defining the model 2. Preparing the data 3. Using the MS Excel regression analysis tool 4. Regression output and model 5. Testing the model 6. How to calculate the RPKs forecast to the year 2015 by using the equation 7. Sensitivity analysis 1. Defining the model The first step in the development of forecasts using regression analyses is to define the variable to be forecast (explained variable) as well as the explanatory variables. The latter variables are those thought to be driving variations in the forecast variable. In the case of air traffic, the explained variable can be the number of passengers, passenger-kilometres flown, freight tonnes or freight tonne-kilometres flown. The explanatory variables can be Income as measured by gross domestic product (GDP), average prices as measured by yields (or unit revenues), exports, imports, etc. For specific routes, other variables may include frequency of flights, distance, etc. Once the explained and the explanatory variables are defined, the model functional form needs to be chosen. Functional forms can be linear, logarithmic, exponential, power, etc. The choice of the functional form is determined by the nature of the relationship between the explained and the explanatory variables. For this exercise, the forecast (explained) variable is world revenue passenger-kilometres (RPK) carried out by scheduled airlines. The explanatory variables are GDP and yield (YLD). The functional form is a constant elasticity model: RPK = k * GDP α * yield β Where k, α and β are constants. α is a measure of the elasticity of demand to GDP and β is a measure of the elasticity of demand to price. In order to estimate the parameters of this model (k, α and β) using linear regression, the following transformation needs to be performed: Ln (RPK) = Ln(k * GDP α * yield β ) = Ln(k) + αln(gdp) + βln(yield) 2. Preparing the data The historic data for the variables concerned can be yearly, quarterly or monthly depending upon the type of forecast required. Generally, the longer the historic data the better the result. Data should be treated for missing or invalid values.

5 - 5 - Basic data Table 1 shows the basic data for RPK, GDP and YLD for the 1975 to 2005 period. Table 1 also shows the year-to-year growth in per cent for these variables. The formula to calculate the per cent growth from one year to another in a variable is given below: Absolute value for year 2 Growth rate (in per cent) = ( ) * 100 Absolute value for year 1 In general, the effect of GDP on the RPK is direct, meaning that if GDP goes up then the traffic will follow the same trend; however, the effect of YLD on RPK is inverse, meaning that an increase (or positive) in YLD will affect the RPK growth adversely.

6 - 6 - Basic data World scheduled revenue parssenger-kilometres World real GDP growth (%) and Index World current yield in US Cents and growth (%) TABLE 1 World Real GDP Real GDP Current YLD YLD Year RPK RPK Index Growth US Cents Change (%) (%) (%)

7 - 7 - Treating data for input Data in terms of absolute values or growth rates often comes in current monetary values which includes the effect of inflation. Most econometric modeling, however, is performed on variables expressed in real (constant) terms i.e. excluding the effect of inflation 1. Table 2 shows GDP data in real terms and YLD data in both current and real terms. As previously explained, GDP index is drawn from GDP growth rates which are already available in real terms, whereas the YLD variable, available in current terms, has to be converted into real terms. The conversion procedure requires an additional series to be used as a deflator 2 which could be the consumer price index (CPI) or the producer price index (PPI). A deflator must correspond the same origin as the variable being deflated, e.g., a variable from a country should be deflated by a deflator from the same country. Also, a variable in terms of US dollars should be deflated by a deflator based upon US dollars. Most deflators are available in UN economic publications as well as International Monetary Fund (IMF) publications. There are two ways to convert a variable into real terms: 1) The variable is available in current terms in the form of year-to-year growth rates: The growth rate of the deflator is subtracted from the growth rate of the variable for each year as described b the formula below: Growth (%) of series in real-term for year 1 = (Growth (%) of series in current-term for year 1) (Growth (%) of deflator for the same year).a new year-to-year growth rate variable expressed in real terms is obtained. This variable can be converted into an index by using the procedure described below. This method is applied in Table 2 to calculate the real YLD index. 2) The variable is available in current terms in the form of absolute values: Each value of the variable expressed in current terms, is divided by the value of the deflator for the same year. A new variable expressed in real terms is obtained. Since deflators are often available in per cent growth and to be able to make the division it has to be transformed into an index first. How to make an index from a growth rate series Given the functional form used in this exercise, the use of an index for one or more explanatory variables does not affect the values of the elasticities α and β. The use of indices is sometimes necessary because the data is only available in the form of growth rates and not absolute values. Notes: 1 The overall general upward price movement of goods and services in an economy, usually as measured by the consumer price index (CPI) or the producer price index (PPI). 2 A tool used to convert money into inflation-adjusted money, in order to compare prices over time after factoring out the overall effects of inflation.

8 - 8 - Note that the GDP variable in Table 1 is provided as an index. This is a series built from year-to-year growth rates of GDP. The world GDP data is available in the form of annual per cent growth rates of GDP volume in constant terms. An index must be built from these growth rates in order to achieve consistency with the other variables expressed in volume. The volume of GDP (in constant dollars for example) can also be used if available. To build an index from per cent growth rates, the value of one of the years (the base year) is chosen to be 100 and the values for the rest of the years can be calculated by the formula below: growth rate for year 2 Index value for year 2 = Index value for year 1 * ( ) 100 In Table 2, the YLD is derived from the aggregate of total world passenger revenues in US $ and divided by the corresponding RPKs. The value obtained is in current US cents and to deflate this value, the industrial countries CPI is considered to be the closest deflator.

9 - 9 - TABLE 2 Transform yield in current values into real-term using Industrial Countries' CPI Build a Series or an Index from growth rates (%) REAL World Real GDP Real GDP Current YLD YLD CPI YLD GROWTH (%) REAL YLD Year RPK RPK Index Growth US Cents Change IND.CPI (YLD%-CPI%) Index (%) (%) (%) (%)

10 Input data Table 3 shows a complete set of data ready for input into the regression. The model equation that will be used is of the form: y = k x 1 α * x 2 β where y = traffic (RPKs) x 1 = Income (GDP) x 2 = Price (YLD) k, α and β are constants. The equation above assumes a non-linear relationship between air traffic demand (RPK), income (GDP) and price (YLD). Since we will run a linear regression using Excel, we need to turn the equation above into a linear form by taking the natural logarithm (Ln) of its both sides, resulting in the following form: Ln (RPK) = k + α * Ln(GDP) + β * Ln(YLD) Table 3 shows the Ln values of RPK, GDP and YLD by simply invoking the LN function using the Excel spreadsheet. Note that in addition, average annual growth rates of each variable for the last 20 years have also been calculated using the formula given below: Formula for calculating average annual growth: Value for last year 1 Average annual growth (%) = (( )^( ) 1)*100 Value for first year Number of years A DUMMY variable is also given in Table 3, the purpose of which is to offset the effects of abnormal events leading to drastic changes in the variables (on a yearly basis). This variable only assumes the values of 0 and 1. It should be noted that the ICAO Manual on Air Traffic Forecasting (Doc 8991) explains in detail the techniques and uses of econometric regression analyses. This manual can be purchased through the ICAO website or by writing to ICAO Document Sales Unit.

11 TABLE 3 TAKE NATURAL LOGARITHM OF RPK, REAL GDP AND REAL YLD ADD DUMMY VARIABLE AND RUN THE MODEL CALCULATE AVERAGE ANNUAL GROWTH OF HISTORICAL DATA World Real GDP Real GDP Current YLD YLD CPI REAL YLD Year RPK RPK Index Growth US Cents Change IND.CPI YLD%-CPI% Index =LN(RPK) =LN(GDP) =LN(YLD) (%) (%) (%) (%) RPK GDP YLD DUMMY AVERAGE ANNUAL GROWTH (%)

12 Using the MS Excel regression analysis tool This section provides a step-by-step approach on how to perform linear regression using MS Excel. Step 1: Check if the Analysis Toolpak is installed The Analysis Toolpak is an Excel add-in (add-in: A supplemental program that adds custom commands or custom features to Microsoft Office) program that is available when you install Microsoft Office or Excel. To use it in Excel, however, you need to load it first. 1. On the Tools menu, click Add-Ins. 2. In the Add-Ins available box, select the check box next to Analysis Toolpak, and then click OK. Tip If Analysis Toolpak is not listed, click Browse to locate it. 3. If you see a message that tells you the Analysis Toolpak is not currently installed on your computer, click Yes to install it. 4. Click Tools on the menu bar. When you load the Analysis Toolpak, the Data Analysis command is added to the Tools menu. Step 2: Entering the data The first step is to have the data presented in Table 3 entered into an Excel spreadsheet as shown in Figure 1 below. Figure 1 Step 3: Selecting the Data Analysis option Then, select Tools on the main menu and Data Analysis from the drop-down list as illustrated in Figure 2.

13 Figure 2 Step 4: Select Regression From the list of data analysis options, select Regression and then click OK as shown in Figure 3. Figure 3 Step 5: Select the input data and parameters After selecting Regression, a new window appears on the screen prompting the user to input the data and parameters required for the regression as shown in Figures 4 and 5.

14 Figure 4 The regression window in Figure 5 has four sections: Input, Output options, Residuals and Normal Probability. The first section is described in this step and the second will be described in the next step. The Input section has the following items: Input Y Range: explained variable. Input X Range: explanatory variables. Labels: Allows variable names (labels) to be included in the selection. Constant is Zero : Allows the user to impose the constant (equation parameter k in our case) to be equal to zero. Confidence Level: Allows the selection of the confidence level to apply to the equation parameters in the regression output. By default, the confidence level is 95%.

15 Figure 5 To select the explained variable (RPK), click on the button to the right in front of Input Y Range textbox. The regression window will be minimized allowing you to make the selection of explained variable. In Figure 6, the explained variable RPK is under column N. When the selection is complete, click on the button on the right side of the minimized regression window. Figure 6

16 Figure 7 To select the explanatory variables (GDP and YLD), click on the button to the right in front of Input X Range textbox. The regression window will be minimized allowing you to make the selection of explanatory variables. In Figure 8, the explanatory variables RPK is under columns O and P. When the selection is complete, click on the button on the right side of the minimized regression window. Figure 8 As can be seen on Figure 9, both the Y and X input ranges are selected. If the selection includes the names of the variables, then the check-box in front of the item Labels should be checked. There is no need to impose Constant is Zero and no need to change the default Confidence Level.

17 Figure 9 Step 6: Select Output option The Output parameters section on the regression window has the following items: Output Range: Allows the selection of regression output location on the same spreadsheet as the input data. New Worksheet Ply: Directs the regression output to a new worksheet ply. New Workbook: Directs the regression output to a new workbook. Click on the radio-button to the left of Output Range to select the range and then click on the button to the right in front of Output Range textbox. The regression window will be minimized allowing you to select the output range. All that is required is to put the cursor in the cell where you want the output of the regression to start. In Figure 10, this cell is S7. When the selection is complete, click on the button on the right side of the minimized regression window.

18 Figure 10 Then, the Regression window will re-appear as shown in Figure 11. Figure 11

19 To perform the regression click on OK. The regression output will be produced as shown on Figure 12. Figure 12

20 Regression output and model Figure 12 shows the Summary Output generated by the regression. The definition of each term in this summary is as follow: Residuals: Actual (or observed) values minus the predicted (estimated using the regression equation) values of the explained variable. Multiple R: The multiple correlation coefficient which measures the correlation between the actual (observed) and the predicted (estimated using the regression equation) values of the explained variable. It lies between -1 and +1. R-Square: The Square of the multiple correlation coefficient. It is also called coefficient of determination. It measures the goodness of fit of the regression (i.e. the proportion of the variability in the explained variable that is explained by the regression). R-Square lies between zero and 1 and the closer it is to 1, the higher the goodness of fit. Adjusted R-Square: The R-Square adjusted to take into account the number of explanatory variables in the regression (or the degrees of freedom). The adjusted R-Square can be negative, and is always less than or equal to R-Square. Standard Error: The standard error of the regression (it measures the average error/residual due to each observation, corrected for the number of variables included in the regression). df: Degrees of freedom of the regression (equal to the number of explanatory variables) of the residuals as well as the total (the number of observations minus one). SS: Sum of squares (squared deviations from the mean) of the regression of the residuals as well as the total. MS: Mean square (mean of squared deviations from the mean) of the regression of the residuals as well as the total. It is equal to the corresponding SS divided by the corresponding df. F: The F statistic. A measure to assess the overall significance of the regression. It is equal to the ratio of two mean squares: the mean square of the regression divided by the mean square of the residuals. In order for the regression to be significant, the F-value must be greater than a critical value, determined by the regression degrees of freedom and the F statistical distribution. Significance F: The probability that the regression is not significant which is equivalent to the probability that all coefficients are equal to zero. Intercept: The estimated constant term in the regression equation. It can be interpreted as the mean effect on the explained variable of all the variables excluded from the regression. Coefficients: The estimated coefficients of the explanatory variables in the regression equation. t-stat: A value of the Student t statistic to assess the significance of the coefficient. In order for the coefficients to be significant (not equal to zero) in general, the absolute value of t-stat must be 2 or more.

21 P-value: Probability that the corresponding intercept or coefficient is equal to zero. Upper and Lower Confidence Levels: Lower 95% and Upper 95% means that there is a 95% chance that the coefficient computed by regression falls between these limits. The upper and lower confidence levels for each coefficient must not cross 0, which would imply that there is a possibility that a coefficient could assume the value 0. If you change the confidence level in the Regression window, these values will change. From this summary, the coefficients of the regression are: Constant = GDP = YLD = Dummy = Hence the equation will be: Ln(RPK) = *Ln(GDP) *Ln(YLD) * (Dummy) The R-Square and the Adjusted R-Square are both close to one signalling a high goodness of fit. The Significance F is very close to zero, implying that the regression is highly significant overall. The t-stat of all the coefficients are much higher than 2, except for the intercept, implying that the coefficients of GDP, YLD and the DUMMY are highly significant.

22 Testing the model Table 4 below shows how to test the model for its accuracy and if it can be used for producing forecasts. In order to manually assess the goodness to fit, residual values can be analyzed. Using the regression equation, calculate the predicted values for each year between 1975 and 2005 and compare them with the actual values. How to calculate predicted values for the sample data (between 1975 and 2005) The equation obtained from the regression is: Ln(RPK) = *Ln(GDP) 0.34*Ln(YLD) * (Dummy) By simply plugging in the Ln(GDP) and Ln(YLD) along with Dummy values in the above equation between the years 1975 to 2005, the predicted value for each year can be calculated as shown in Table 4. The actual minus the predicted would be the errors (or the residuals). These errors can then be put in terms of per cent errors. A graph of the residuals will show if they are following any particular pattern. Any cyclical or symmetric patterns would indicate the interference of errors in the computation of Model or what is called Autocorrelation.

23 TABLE 4 USE THE EQUATION FROM MODEL TO CALCULATE RESIDUAL VALUES EXAMINE THE ERROR PERCENT PLOT A GRAPH OF RESIDUALS TO CHECK FOR AUTO-CORRELATION EQUATION: ln (RPK) = *ln(GDP) *ln(YLD) *(DUMMY) YEAR ACTUAL PREDICTED ERROR ERR (%) GRAPH OF RESIDUALS (ERRORS)

24 How to calculate RPKs forecast for the next year using the regression equation The actual RPK data used in the model is until In order to calculate the RPK forecast value for the year 2006, we need to assume growths of GDP and YLD for the same year. Since the Model uses the GDP and YLD index values, the same index values of GPD and YLD will be projected for the year 2006 and will be used for 2006 RPK projection in the Equation. Table 5 on the next page shows the year-toyear projection of RPK based on selected assumptions. An extract of Table 5 is shown below as an example. World Real GDP Real GDP REAL YLD REAL YLD Year RPK RPK Index Growth Index GROWTH (%) DUMMY (%) (%) The assumption for GDP growth for 2006 is 5%, hence the GDP 2005 index grown by 5% (289.9*1.05) is as the 2006 GDP index value. Similarly, the 2006 YLD index value after -1.5 decline is We will use these two values in our equation to calculate the 2006 RPK. Please note that the value for DUMMY is zero in all the forecast years. The regression equation is: ln(rpk) = *ln(GDP) *ln(YLD) *(DUMMY) This is a logarithmic equation, therefore, the antilog function (or exponential function) will be used to get 2006 RPK, as shown below: EXP( *Log(304.4) *Log(46.0) *(0)) As a result of this calculation, the RPK forecast for 2006 is 4037 with a growth of 8.5% over The RPK forecasts for all the years from 2007 to 2015 can be calculated by doing the same operation to GDP and YLD index values and plugging them into the equation, as shown in the next page.

25 TABLE 5 A SAMPLE FORECAST USING FORECAST EQUATION AND ASSUMPTIONS FOR GDP AND YLD 10-YEAR RPK FORECAST ASSUMPTIONS FOR THE NEXT 10 YEARS: GDP = 5 YLD = -1.5 EQUATION: ln (RPK) = *ln(GDP) *ln(YLD) *(DUMMY) World Real GDP Real GDP REAL YLD REAL YLD Year RPK RPK Index Growth Index GROWTH (%) DUMMY (%) (%) Average Annual Growth (%) PLOT A GRAPH SHOWING HISTORICAL AND FORECAST TRENDS YEAR RPK

26 Sensitivity analysis To obtain the RPK forecast for 2015, an average annual growth rate (%) is assumed for GDP and YLD for the forecast years covering 2005 to 2015 period. Table 6 on the next page shows RPKs calculated for the year 2015 with various YLD and GDP assumptions. It also shows the annual average growth rates between 2005 and 2015 for each set of assumptions. An extract of this table is produced below as an example: GDP ======> AVERAGE ANNUAL GROWTHS, ASSUMPTIONS 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% YLD ASSUMPTIONS RPK's in % Growth > 2.8% 3.5% 4.1% 4.7% 5.4% 6.0% Several combinations of high and low growth rates for the GDP and YLD can be assumed for the forecast period. For example, in one of the options in the sensitivity analysis table above, the assumed growth for the GDP until 2015 is 2.5% and for the YLD it is 1.5%. The GDP and YLD indices for 2005 are then projected by applying these growth rates until 2015, which is a period of 10 years 1 : GDP index value for 2015 = 289.9*1.025^10 = YLD index value for 2015 = 46.7*1.015^10 = 54.2 and plugged in the equation as shown below: EXP( *Log(371.1) *Log(54.2) *(0)) = 4915 The RPK value for the year 2015 obtained from the equation is 4915, which represents an annual average growth of 2.5% between 2005 and Similarly, various low and high RPK forecasts for 2015 can be calculated depending upon different low and high assumptions of GDP and YLD as shown in the table on next page. 1 Projected annual growth = (present year value)*(1 + %growth/100)^(number of years)

27 TABLE 6 DEVELOP A "SENSITIVITY ANALYSIS" MATRIX USING THE FORECAST EQUATION ASSUME HIGH AND LOW GROWTH RATES OF GDP AND YLD FOR THE FORECAST PERIOD EQUATION: ln (RPK) = *ln(GDP) *ln(YLD) *(DUMMY) ACTUAL RPK 2005= 3720 GDP ======> AVERAGE ANNUAL GROWTHS, ASSUMPTIONS 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% YLD ASSUMPTIONS RPK's in % Growth > 2.8% 3.5% 4.1% 4.7% 5.4% 6.0% 1.0% AVERAGE Growth > 3.0% 3.6% 4.3% 4.9% 5.6% 6.2% ANNUAL GROWTH, 0.0% Growth > 3.3% 4.0% 4.6% 5.3% 5.9% 6.6% PERIOD -1.0% Growth > 3.7% 4.3% 5.0% 5.6% 6.3% 6.9% -1.5% Growth > 3.9% 4.5% 5.2% 5.8% 6.5% 7.1% -2.0% Growth > 4.1% 4.7% 5.4% 6.0% 6.7% 7.3% END