# Linear regression

The other type of forecasting methods is linear regression. The regression model proposed by Wickham (1995) consists of dependent variable Yt, which forecasted number of rooms booked in period t and independent variable Y(t-n), which states for number of rooms booked n days before arrival. The regression is described by following equation:

where beta1 and beta2 and are parameters of the regression. The least squared error method is recommended to estimate parameters. This regression model can be expended by more independent variables. Then, equation takes the form:

where Y(t-n),…,Y(t-n-k) are number of reservations n days before arrival date. The more advance form of regression model proposed Phumchusri and Mongkolkul (2012):

where Yt is the forecast of bookings on day , Bk is number of bookings days prior to day , Mon…Sun are dummy variables which are equal to 1 when the day of stay is Mon, Tue, … , Sun respectively, beta0 is constant and beta1,…,beta16 are coefficients of described parameters. One of the methods Weatherford and Kimes (2003) used in their comparison study was a logarithmic linear regression

where Bk is number to bookings on day k and beta0 and beta1 are parameters. The other important thing is method, which we will use to estimate the parameters of the regression. The most popular and giving the best results is ordinary least squares (OLS) methods, however this method requires numerous of assumptions which all of them need to be fulfilled in order to find the most efficient parameters (most efficient means sum of squared residuals is minimized). Those assumptions are:

• linearity of parameters (the dependent variable may be calculated as a function of specific set of independent variables plus an error term),
• randomness of observations (the sample consist of n observations drawn from population; the number of observations is greater than number of parameters which need to be estimated n > k; the independent variables are nonstochastic),
• zero conditional mean (the mean of errors term has an expected value of zero),
• no perfect colinearity (there is no exact linear relation between independent variables),
• homoscedasticity (the error terms have the same variance and are not correlated with each other).

There are several methods presented in table below, which test the OLS assumptions. The reason why we use this estimation method, even if it has so strong assumptions is fact that it creates the best fitting models because it minimizes the errors (based on Gauss-Markov Theorem).

 Assumption Test Disturbance distribution Normality Jarque-Bera Heteroscedasticity White, Goldfeld-Quandt, Breusch-Pagan, ARCH Autocorrelation Durbin-Watson, Breusch-Godfrey, Box-Pierce, Lijung-Box Specification and functional form AIC, RESET Stability Chow, CUSUM

The application of this method on our Regression Dataset was undertook in Gretl and this process is explained here: Regression – explanation

Example

Linear Regression is very efficient as well in our calculations as in following example. The demand here comes from real hotel and it is very unstable (therefore, other methods were inefficient). Compare to other methods where MAPE accounted to not less than 45%; linear regression enabled to achieve MAPE=13%. Model 63 has structural form

References

1. Phumchusri, D., Mongkolkul, J. (2012) Hotel Room Demand via Observed Reservation Information. Proceedings of the Asia Pacific Industrial Engineering & Management Systems Conference 2012, pp. 1978-1985
2. Weatherford, L.R. & Kimes, S.E. (2003). A comparison of forecasting methods for hotel revenue management. International  Journal of Forecasting, vol. 19, no. 3, pp. 401-415.
3. Wickham, R. (1995) Evaluation of forecasting techniques for short-term demand of air transportation. Master’s thesis, Massachusetts Institute of Technology
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