Slides sales-forecasting-session1-web

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Slides sales-forecasting-session1-web

  1. 1. Arthur CHARPENTIER - Sales forecasting. Sales forecasting # 1 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1
  2. 2. Arthur CHARPENTIER - Sales forecasting. Agenda Qualitative and quantitative methods, a very general introduction • Series decomposition • Short versus long term forecasting • Regression techniques Regression and econometric methods • Box & Jenkins ARIMA time series method • Forecasting with ARIMA series Practical issues: forecasting with MSExcel 2
  3. 3. Arthur CHARPENTIER - Sales forecasting. Somes references Major reference for this short course, Pindyck, R.S. & Rubinfeld, D.L. (1997). Econometric models and economic forecasts. Mc Graw Hill. “A forecast is a quantitative estimate about the likelihood of future events which is developed on the basis of past and current information”. 3
  4. 4. Arthur CHARPENTIER - Sales forecasting. Forecasting challenges ? “With over 50 foreign cars already on sale here, the Japanese auto industry isn’t likely to carve out a big slice of the U.S. market”. - Business Week, 1958 “I think there is a world market for maybe five computers”. - Thomas J. Watson, 1943, Chairman of the Board of IBM “640K ought to be enough for anybody”. - Bill Gates, 1981 “Stocks have reached what looks like a permanently high plateau”. - Irving Fisher, Professor of Economics, Yale University, October 16, 1929. 4
  5. 5. Arthur CHARPENTIER - Sales forecasting. Challenge: use MSExcel (only) to build a forecast model MSExcel is not a statistical software. Specific softwares can be used, e.g. SAS, Gauss, RATS, EViews, SPlus, or more recently, R (which is the free statistical software). 5
  6. 6. Arthur CHARPENTIER - Sales forecasting. Macro versus micro ? Macroeconomic Forecasting is related to the prediction of aggregate economic behavior, e.g. GDP, Unemployment, Interest Rates, Exports, Imports, Government Spending, etc. It is a very difficult exercice, which appears frequently in the media. 6
  7. 7. Arthur CHARPENTIER - Sales forecasting. −4−20246810 American Express University of North Carolina Goldman Sachs PNC Financial Kudlow & co Figure 1: Economic growth forecasts, from Wall Street Journal, Sept. 12, 2002, Q4 2002, Q1 2003 and Q2 2003. 7
  8. 8. Arthur CHARPENTIER - Sales forecasting. Macro versus micro ? Microeconomic Forecasting is related to the prediction of firm sales, industry sales, product sales, prices, costs... Usually more accurate, and applicable to business manager... Problem is that human behavior is not always rational: there is always unpredictable uncertainty. 8
  9. 9. Arthur CHARPENTIER - Sales forecasting. Short versus long term? 0 50 100 150 200 250 −4 −2 0 2 4 6 8 0 50 100 150 200 250 −5 0 5 0 50 100 150 200 250 −20 −10 0 10 Figure 2: Forecasting a time series, with different models. 9
  10. 10. Arthur CHARPENTIER - Sales forecasting. Short versus long term? 160 180 200 220 240 −4 −2 0 2 4 6 8 160 180 200 220 240 −4 −2 0 2 4 6 8 160 180 200 220 240 −4 −2 0 2 4 6 8 Figure 3: Forecasting a time series, with different models. 10
  11. 11. Arthur CHARPENTIER - Sales forecasting. Short versus long term? q qq qq q qq qq q qq qq qq q q q q q qq qq q q qqqqq qqqqqqqqqqq qq qqq q q qqqqq qq q q q qqq qqqq q q q qq q q qq qqqqqq q q qq q qqq q q q qqqq qqq qqq qqq qq q qq q qqq q qq q q q qq q q q q qq q q q qqq qqqqq qq qq q q qqqqq qqq qq q qq q qqqqqq q q qqq q q qq qq qq qq q q q q q q q qqq qq q q qqqq qq q qq qqqq q qqq q q q qqq q qqqq q qq q q q qq qq qq q qq q qqq q qq qq qqq qq qq qqq q qq qqqqq q qq q qqq q qq q q qq q q q q qq qqqq q q q qqq q qq q q q q q qq qq qqq qqq qq q q qqq q q q qq qqqqq qq qq qqq q qq qq qq q qq qqq q qq qqqq qqqq q q qq q q q q q q qqq qqqq qq qq q qqq q q qqq q q qq qq q qqqq qq qqqqqqqq q q q qq q qq q qqq qqq q qq q q qqq qqq q q q q q qq q q q q q qqqqq q qqq q qqqqqqqqq q q qqq qq q q qq qqqqq q q q q qq qqq q q q qq q qq q qq q q q q q qq q q qq q q q qq q q qqqqq qq q q q q qq q q q q qqq q q qqqq qqq qq q qq q q q q qqq q qqq q q qq q qq qq q q q q qq q q q qq q q qq qq qq q q q qq q q qq qq q qq q q qq q q q q qq qq qq 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qq q qq q q q qq q q qq qqq q q q q q qq qq q q q q q qq q qq q q q q q qqq q q q qq qq q q q qq qq q q qq q q q q q q q q q qqqq q q q q q q q qqq qq q q q q qq q q qq q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q qqq q q q q qq q q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q q q qq q q q q q q qq qq q q q q q qq qq q q q q qqq q q qq q q q q q qq qq qq qq qq q qq q q qqq q q q q qq qq qq q q q qq q qq q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q qqq q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q qq q qqq q q q qq q q q q q qq q q q q q q q q q q q q q qqq q q q qq qq q q q qqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q q q q q q q q q q qq q q q qq q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq qqq q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq qq q q q qq q qq qq q q q q q q q qq q q qq q q qq q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q qq q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q qqq q q q q q q q q q q q q q q q q q qq q q q q q q q q q qqq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq qq q q q q q q q q qqq q q q q q q q q q q q q q qq q q q q q qq q q q qq q q q q q q q q q q q q qq q q q q q q q qq q qqq q q q q q q qq q q q q q q q q q q q qqq q q q q q q q q qqq q q qq q q qq q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q qq qq q q q q q q q qq q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q qq q q q qq q q q qq q q q q q qq q q q q q q q q q q qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q qqq q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q qq qq q qq qq q qqqqq q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q qq q q q q q q qq q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q qq q qq q q q q q q q q q q qqqq qqq qq q qq q q q q q q q q qq q q qqq q q qq q q q qqq q qq q q qqq q q q qq q q q q qq qq q q q q q q q qq q q q q q q q qq q q q q q qq q q qq q q qq q q q qq q q q q qqq q q q q qq q q q q q qq q q q q q q q q q q q qq q qq q q q q qq q q q qqq q q q q q q q q qq q q qqq q qqqq qq q q q q q q q q q q qq qq q q q q q q qqq q q q q q qq q q q q q q q q qq qq q q q q q qq q q q qq q q q q qq q qqq qq q qq q q q q qqq q q q q q q q q qq q qq q q qqq q q q qq q q q q qqqq q qq q q q q qq qqqqqq q q qq q q q qq q qq q q q q q q q q q q q q q qq qqq q q q q q q q q q qq qq q q q q q q qq qq q q qq q q q qq q q q q q q q qqq qq q q q q qq q q q q q qqq q qq q q qq q q q q q qq q q q qq q q q q q q q q qqqq q q q q qq q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q qq qqq q q q q q q qq q q q qq q qq q qq q q qq q q qqq q q qq q q qq q q q q q q q q q q qq q q qq q q q q q q q q qq q qq q qqq qq q qq q q q q q q q q q q q qq q q q q q qq qq qq qq q q q qq q qq q q q q q q q q qqqq qq qq q q q q q q q q q qq qq q q q q q q q q q q q q q q qq q qq q qq qqq q qq q q q q q q qqq q qq q qq q q q qqqqq q q qq q q qq q q q qqqqqq q q q q q q q q q qq q q q q q q qq q qqqq q q q q q q q q q q The Nasdaq index, 1971−2007 1970 1980 1990 2000 −0.10−0.050.000.050.10 010002000300040005000 Dailylogreturn LeveloftheNasdaqindex Figure 4: Forecasting financial time series. 11
  12. 12. Arthur CHARPENTIER - Sales forecasting. Series decomposition Decomposition assumes that the data consist of data = pattern + error Where the pattern is made of trend, cycle, and seasonality. General representation is Xt = f(St, Dt, Ct, εt) where • Xt denotes the time series value at time t, • St denotes the seasonal component at time t, i.e. seasonal effect, • Dt denotes the trend component at time t, i.e. secular trend, • Ct denotes the cycle component at time t, i.e. cyclical variation, • εt denotes the error component at time t, i.e. random fluctuations, 12
  13. 13. Arthur CHARPENTIER - Sales forecasting. Series decomposition The secular trends are long-run trends that cause changes in an economic data series, three different patterns can be distinguished, • linear trend, Yt = α + βt • constant rate of growth trend, Yt = Y0(1 + γ)t • declining rate of growth trend, Yt = exp(α − β/t) For the linear trend, adjustment can be obtained, introducing breaks for instance. For constant rate of growth trend, note that in that case log Yt = log Y0 + log(1 + γ) · t, which is a linear model on the logarithm of the serie. 13
  14. 14. Arthur CHARPENTIER - Sales forecasting. Series decomposition For those two models, standard regression techniques can be used. For declining rate of growth trend, log Yt = α − β/t, which is sometimes called semilog regression model. The cyclical variations are major expansions and contractions in an economic series that are usually greater than a year in duration The seasonal effect cause variation during a year, that tend to be more or less consistent from year to year, From an econometric point of view, a seasonal effect is obtained using dummy variables. E.g for quaterly data, Yt = α + βt + γ1∆1,t + γ2∆2,t + γ3∆3,t + γ4∆4,t where ∆i,t is an indicator series, being equal to 1 when t is in the ith quarter, and 0 if not. The random fluctuations cannot be predicted. 14
  15. 15. Arthur CHARPENTIER - Sales forecasting. qq qq q q q q q q q q qq q q q q q q q qq qqq q q q q qqqq qqq q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q qq q q q q qq q qq 0 50 100 150 200 405060708090 Figure 5: Standard time series model, Xt. 15
  16. 16. Arthur CHARPENTIER - Sales forecasting. qq qq q q q q q q q q qq q q q q q q q qq qqq q q q q qqqq qqq q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q qq q q q q qq q qq 0 50 100 150 200 405060708090 Figure 6: Standard time series model, the linear trend component. 16
  17. 17. Arthur CHARPENTIER - Sales forecasting. qq q q q q q q q q q q qq q q q q q q q qq qq q q q q q qq qq qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q qq q q q 0 50 100 150 200 −20−1001020 Figure 7: Removing the linear trend component Xt − Dt. 17
  18. 18. Arthur CHARPENTIER - Sales forecasting. qq q q q q q q q q q q qq q q q q q q q qq qq q q q q q qq qq qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q qq q q q 0 50 100 150 200 −20−1001020 Figure 8: Standard time series model, detecting the cycle on Xt − Dt. 18
  19. 19. Arthur CHARPENTIER - Sales forecasting. qq qq q q q q q q q q qq q q q q q q q qq qqq q q q q qqqq qqq q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q qq q q q q qq q qq 0 50 100 150 200 405060708090 Figure 9: Standard time series model, Xt. 19
  20. 20. Arthur CHARPENTIER - Sales forecasting. qq q q q q q q q q q q q q q q q q q q q qq qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q 0 50 100 150 200 −10−50510 Figure 10: Removing linear trend and seasonal component Xt − Dt − St. 20
  21. 21. Arthur CHARPENTIER - Sales forecasting. Exogeneous versus endogenous variables Model Xt = f(St, Dt, Ct, εt, Zt) can contain on exogeneous variables Z, so that • St, the seasonal component at time t, can be predicted, i.e. ST +1, ST +2, · · · , ST +h • Dt, the trend component at time t, can be predicted, i.e. DT +1, DT +2, · · · , DT +h • Ct, the cycle component at time t, can be predicted, i.e. CT +1, CT +2, · · · , CT +h • Zt, the exogeneous variables at time t, can be predicted, i.e. ZT +1, ZT +2, · · · , ZT +h • but εt, the error component cannot be predicted 21
  22. 22. Arthur CHARPENTIER - Sales forecasting. Exogeneous versus endogenous variables Like in classical regression models: try to find a model Yi = Xiβ + εi which the highest prediction value. Classical ideas in econometrics: compare Yi and Yi, which should be as closed as possible. E.g. minimize n i=1 (Yi − Yi)2 , which is the sum of squared errors, and can be related to the R2 , or MSE, or RMSE. When dealing with time series, it is possible to add an endogeneous component. Endogeneous variables are those that the model seeks to explain via the solution of the system of equations. The general model is then Xt = f(St, Dt, Ct, εt, Zt, Xt−1, Xt−2, ..., Zt−1, ..., εt−1, ...) 22
  23. 23. Arthur CHARPENTIER - Sales forecasting. Comparing forecast models In order to evaluate the accuracy - or reliability - of forecasting models, the R2 has been seen as a good measure in regression analysis,but the standard is the root mean square error (RMSE), i.e. RMSE = 1 n n i=1 (Yi − Yi)2 where is a good measure of the goodness of fit. The smaller the value of the RMSE, the greater the accurary of a forecasting model. 23
  24. 24. Arthur CHARPENTIER - Sales forecasting. q q ESTIMATION PERIOD EX−POST FORECAST PERIOD EX−ANTE FORECAST PERIOD Figure 11: Estimation period, ex-ante and ex-post forecasting periods. 24
  25. 25. Arthur CHARPENTIER - Sales forecasting. Regression model Consider the following regression model, Yi = Xiβ + εi. Call: lm(formula = weight ~ groupCtl+ groupTrt - 1) Residuals: Min 1Q Median 3Q Max -1.0710 -0.4938 0.0685 0.2462 1.3690 Coefficients: Estimate Std. Error t value Pr(>|t|) groupCtl 5.0320 0.2202 22.85 9.55e-15 *** groupTrt 4.6610 0.2202 21.16 3.62e-14 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.6964 on 18 degrees of freedom Multiple R-Squared: 0.9818, Adjusted R-squared: 0.9798 F-statistic: 485.1 on 2 and 18 DF, p-value: < 2.2e-16 25
  26. 26. Arthur CHARPENTIER - Sales forecasting. Lest square estimation Parameters are estimated using ordinary least squares techniques, i.e. β = (X X)−1 X Y . E(β) = β. q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q 5 10 15 20 25 020406080100120 car speed distance Linear regression, distance versus speed Figure 12: Least square regression, Y = a + bX. 26
  27. 27. Arthur CHARPENTIER - Sales forecasting. Lest square estimation Parameters are estimated using ordinary least squares techniques, i.e. β = (X X)−1 X Y . E(β) = β. q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q 5 10 15 20 25 020406080100120 car speed distance Linear regression, speed versus distance Figure 13: Least square regression, X = c + dY . 27
  28. 28. Arthur CHARPENTIER - Sales forecasting. Lest square estimation Assuming ε ∼ N(0, σ2 ), then V (β) = (X X)−1 σ2 . The variance of residuals σ2 can be estimated using ε ε/(n − k − 1). It is possible to test H0 : βi = 0, then βi/σ (X X)−1 i,i has a Student t distribution under H0, with n − k − 1 degrees of freedom. The p-value corresponding to the power of the t-test, i.e. 1- probability of second type error. The confidence interval for βi can be obtained easilty as βi − tn−k(1 − α/2)σ [(X X)−1]i,i; βi + tn−k(1 − α/2)σ [(X X)−1]i,i where tn−k(1 − α/2) stands for the (1 − α/2) quantile of the t distribution with n − k degrees of freedom. 28
  29. 29. Arthur CHARPENTIER - Sales forecasting. Lest square estimation 3.5 4.0 4.5 5.0 5.5 −0.010.010.020.030.04 Endemics Area q −0.15 −0.10 −0.05 0.00 0.05 −0.010.010.020.030.04 Elevation Area q 29
  30. 30. Arthur CHARPENTIER - Sales forecasting. Lest square estimation The R2 is the correlation coefficient between series {Y1, · · · , Yn} and {Y1, · · · , Yn}, where Yi = Xiβ. It can be interpreted as the ratio of the variance explained by regression, and total variance. The adjusted R2 , called R 2 , is defined as R 2 = (n − 1)R2 − k n − k = 1 − n − 1 n − k − 1 (1 − R2 ). Assume that residuals are N(0, σ2 ), then Y ∼ N(Xβ, σ2 I), and thus, it is possible to use maximum likelihood technique, log L(β, σ|X, Y ) = − n 2 log(2π) − n 2 log(σ2 ) − (Y − Xβ) (Y − Xβ) 2σ2 Akake criteria (AIC) and Schwarz criteria (SBC) can be used to choose a model. AIC = −2 log L + 2k and SBC = −2 log L + k log n 30
  31. 31. Arthur CHARPENTIER - Sales forecasting. Lest square estimation Fisher’s statistics can be used to test globally the significance of the regression, i.e. H0 : β = 0, defined as F = n − k k − 1 R2 1 − R2 . Additional tests can be run, e.g. to test normality of residuals, such as Jarque-Berra statistics, defined as BJ = n 6 sk2 + n 24 [κ − 3]2 , where sk denotes the empirical skewness, and κ the empirical kurtosis. Under assumption H0 of normality, BJ ∼2 (2). 31
  32. 32. Arthur CHARPENTIER - Sales forecasting. Residual in linear regression 1 2 3 4 5 6 −3−2−10123 Fitted values Residuals q q q q q q q q q q qq q q lm(Y ~ X1 + X2) Residuals vs Fitted 2 5 4 q q q q q q q q q q q q q q −1 0 1 −1012 Theoretical Quantiles Standardizedresiduals lm(Y ~ X1 + X2) Normal Q−Q 2 5 1 32
  33. 33. Arthur CHARPENTIER - Sales forecasting. Prediction in the linear model Given a new observation x0, the predicted response is x0β. Note that the associated variance is V ar(x0β) = x0(X X)−1 x0σ2 . Since the future observation should be x0β+ε (where ε is unknown, but yield additional uncertainty), the confidence interval for this predicted value can be computed as βi − tn−k(1 − α/2)σ 1+x0(X X)−1x0; βi + tn−k(1 − α/2)σ 1+x0(X X)−1x0 where again tn−k(1 − α/2) stands for the (1 − α/2) quantile of the t distribution with n − k degrees of freedom. Remark Recall that this is rather different compared with the confidence interval for the mean response, given x0, which is βi − tn−k(1 − α/2)σ x0(X X)−1x0; βi + tn−k(1 − α/2)σ x0(X X)−1x0 33
  34. 34. Arthur CHARPENTIER - Sales forecasting. Prediction in the linear model q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q 5 10 15 20 25 020406080100120 car speed distance Confidence and prediction bands q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q 5 10 15 20 25 020406080100120 car speed distance Confidence and prediction bands 34
  35. 35. Arthur CHARPENTIER - Sales forecasting. Regression, basics on statistical regression techniques Remark statistical uncertainty and parameter uncertainty. Consider i.i.d. observations X1, lcdot, Xn from a N(µ, σ) distribution, where µ is unknown and should be estimated. Step 1: in case σ is known. The natural estimate of unkown µ is µ = 1 n n i=1 Xi, and the 95% confidence interval is µ + u2.5% σ √ n ; µ + u97.5% σ √ n where u2.5% = −1.9645 and u97.5% = 1.9645. Both are quantiles of the N(0, 1) distribution. 35
  36. 36. Arthur CHARPENTIER - Sales forecasting. Regression, basics on statistical regression techniques Step 2: in case σ is unknown. The natural estimate of unkown µ is still µ = 1 n n i=1 Xi, and the 95% confidence interval is µ + t2.5% σ √ n ; µ + t97.5% σ √ n The following table gives values of t2.5% and t97.5% for different values of n. 36
  37. 37. Arthur CHARPENTIER - Sales forecasting. n t2.5% t97.5% n t2.5% t97.5% 5 -2.570582 2.570582 30 -2.042272 2.042272 10 -2.228139 2.228139 40 -2.021075 2.021075 15 -2.131450 2.131450 50 -2.008559 2.008559 20 -2.085963 2.085963 100 -1.983972 1.983972 25 -2.059539 2.059539 200 -1.971896 1.971896 Table 1: Quantiles of the t distribution for different values of n. This information is embodied in the form of a model - a single equation structural model, a multiequation model, or a time series model By extrapolating the models beyond the period over which they are estimated ,we get forecasts about future events. 37
  38. 38. Arthur CHARPENTIER - Sales forecasting. Regression model for time series Consider the following regression model, Yt = α + βXt + εt where εt ∼ N(0, σ2 ). Step 1: in case α and β are known, Given a known value XT +1, and if α and β are known, then YT +1 = E(YT +1) = α + βXT +1 This yields a forecast error, εT +1 = YT +1 − YT +1. This error has two properties • the forecast should be unbiased E(εT +1) = 0 • the forecast error variance is constant V (εT +1) = E(ε2 T +1) = σ2 . 38
  39. 39. Arthur CHARPENTIER - Sales forecasting. Regression model for time series Step 2: in case α and β are unknown, The best forecast for YT +1 is then determined from a simple two-stage procedure, • estimate parameters of the linear equation using ordinary least squares • set YT +1 = α + βXT +1 Thus, the forecast error is then εT +1 = YT +1 − YT +1 = (α − α) + (β − β)XT +1 − εT +1 Thus, there are two sources of error: • the additive error term εT +1 • the random nature of statistical estimation 39
  40. 40. Arthur CHARPENTIER - Sales forecasting. Figure 14: Forecasting techniques, problem of uncertainty related to parameter estimation. 40
  41. 41. Arthur CHARPENTIER - Sales forecasting. Regression model for time series Consider the following regression model Goal of ordinay least squares, minimize N I=1(Yi − Yi)2 where Y = α + βX. Then β = n XiYi − Xi Yi n X2 i − ( Xi) 2 and α = Yi n − β · Xi n = Y − βX The least square slope can be writen β = (Xi − X)(Yi − Y ) (Xi − X)2 V (εT +1) = V (α) + 2XT +1cov(α, β) + X2 T +1V (β) + σ2 41
  42. 42. Arthur CHARPENTIER - Sales forecasting. Regression model for time series under the assumption of the linear model, i.e. • there exists a linear relationship between X and Y , Y = α + βX, • the Xi’s are nonrandom variables, • the errors have zero expected value, E(ε) = 0, • the errors have constant variance, V (ε) = σ2 , • the errors are independent, • the errors are normally distributed. 42
  43. 43. Arthur CHARPENTIER - Sales forecasting. Regression model and Gauss-Markov theorem Under the 5 first assumptions, the estimators α and β are the best (most efficient) linear unbiased estimator of α and β, in the sense that they have minimum variance, of all linear unbiased estimators (i.e. BLUE, best linear unbiased estimators). The two estimators are further asymptotically normal, √ n(β − β)→N 0, n · σ2 (Xi − X)2 and √ n(α − α)→N 0, σ2 X2 i (Xi − X)2 . The asymptotic variances of α and β can be estimated as V (β) = σ2 (Xi − X)2 and V (α) = σ2 n (Xi − X)2 while the covariance is cov(α, β) = −Xσ2 (Xi − X)2 . 43
  44. 44. Arthur CHARPENTIER - Sales forecasting. Regression model and Gauss-Markov theorem Thus, if σ denotes the standard deviation of εT +1, the standard deviation s of εT +1 can be estimated as s2 = σ 1+ 1 T + (XT +1 − X)2 (Xi − X)2 > σ. 44
  45. 45. Arthur CHARPENTIER - Sales forecasting. RMSE (root mean square error) and Theil’s inequality Recall that the root mean square error (RMSE), i.e. RMSE = 1 n n i=1 (Yi − Yi)2 Another useful statistic is Theil inequality coefficient defined as U = 1 T n i=1 (Yi − Yi)2 1 T n i=1 Y 2 i + 1 T n i=1 Y 2 i From this normalization U always fall between 0 and 1. U = 0 is a perfect fit, while U = 1 means that the predictive performance is as bad as it could possibly be. 45
  46. 46. Arthur CHARPENTIER - Sales forecasting. Step 3, assume that α, β and XT +1 are unknown, but that XT +1 = XT +1 + uT +1, where uT +1 ∼ N(0, σ2 u). The two errors are uncorrelated. Here, the error of forecast is εT +1 = YT +1 − YT +1 = (α − α) + (β − β)XT +1 − εT +1 It can be proved (easily) that E(εT +1) = 0. But its variance is slightly more complecated to derive V (εT +1) = V (α) + 2XT +1cov(α, β) + (X2 T +1+σ2 u)V (β) + σ2 +β2 σ2 u And therefore, the forecast error variance is then s2 = σ 1 + 1 T + (XT +1 − X)2 + σ2 u (Xi − X)2 + β2 σ2 u > σ2 , which,again, increases the forecast error. 46
  47. 47. Arthur CHARPENTIER - Sales forecasting. To go further, multiple regression model In the multiple regression model, Y = Xβ + ε, in which Y =        Y1 Y2 ... Yn        ,X =        X1,1 X2,1 ... Xk,1 X1,2 X2,2 ... Xk,2 ... ... ... X1,n X2,n ... Xk,n        ,β =        β1 β2 ... βK        ,ε =        ε1 ε2 ... εn        • there exists a textcolorbluelinear relationship between X1, , Xk and Y , Y = α + β1X1 + +βkXk, • the Xi’s are nonrandom variables, and moreover, there are no exact linear relationship between two and more independent variables, • the errors have zero expected value, E(ε = 0, • the errors have constant variance, var(ε) = σ2 , • the errors are independent, 47
  48. 48. Arthur CHARPENTIER - Sales forecasting. • the errors are normally distributed. The new assumption here is that “there are no exact linear relationship between two and more independent variables”. If such a relationship exists, variables are perfectly collinear, i.e. perfect collinearity. From a statistical point of view, multicollinearity occures when two variables are closely related. This might occur e.g. between two series {X2, X3, · · · , XT } and {X1, X2, · · · , XT −1} with strong autocorrelation. 48
  49. 49. Arthur CHARPENTIER - Sales forecasting. To go further, forecasting with serial correlated errors In previous model, errors were homoscedastic. A more general model is obtained when errors are heteroscedastic, i.e. non-constant variance. Goldfeld-Quandt test can be performed. An alternative is to assume serial correlation. Cochrane-Orcutt or Hildreth-Lu procedures can be performed. Consider the following regression model, Yt = α + βXt + εt where εt = ρεt − 1 + ηt with −1 ≤ ρ ≤ +1 and ηt ∼ N(0, σ2 ). Step 1, assume that α, β and ρ are known. YT +1 = α + βXT +1 + εT +1 = α + βXT +1 + ρεT assuming that εT +1 = ρεT . Recursively, εT +2 = ρεT +1 = ρ2 εT 49
  50. 50. Arthur CHARPENTIER - Sales forecasting. εT +3 = ρεT +2 = ρ3 εT εT +h = ρεT +h−1 = ρh εT Since |ρ| < 1, ρh approaches 0 as h gets arbitrary large. Hence, the information provided by serial correlation becomes less and less usefull. YT +1 = α(1 − ρ) + βXT +1 + ρ(YT − βXT ) Since YT = α + βXT + εT , then YT +1 = α + βXT +1 + ρεT Thus, the forecast error is then εT = YT − YT = ρεT − εT +1 50
  51. 51. Arthur CHARPENTIER - Sales forecasting. To go further, using lag models We have mentioned earlier that when dealing with time series, it was possible not only to consider the linear regression of Yt on Xt, but to consider lagged variates • either Xt−1, Xt−2, Xt−2, ...etc, • or Yt−1, Yt−2, Yt−2, ...etc, First, we will focuse on adding lagged explanatory exogneous variable, i.e. models such as Yt = α + β0Xt + β1Xt−1 + β2Xt−2 + · · · + βhXt−h + · · · + εt. Remark In a very general setting Xt can be a random vector in Rk . 51
  52. 52. Arthur CHARPENTIER - Sales forecasting. To go further, a geometric lag model Assume that weights of the lagged explanatory variables are all positive and decline geometrically with time, Yt = α + β Xt + ωXt−1 + ω2 Xt−2 + ω3 Xt−3 + · · · + ωh Xt−h + · · · + εt, with 0 < ω < 1. Note that Yt−1 = α+β Xt−1 + ωXt−2 + ω2 Xt−3 + ω3 Xt−4 + · · · + ωh Xt−h−1 + · · · +εt−1, so that Yt − ωYt−1 = α(1 − ω) + βXt + ηt where ηt = εt − ωεt−1. Rewriting Yt = α(1 − ω) + ωYt−1 + βXt + ηt. 52
  53. 53. Arthur CHARPENTIER - Sales forecasting. To go further, a geometric lag model This would be called single-equation autoregressive model, with a single lagged dependent variable. The presence of a lagged dependent variable in the model causes ordinary least-squares parameter estimates to be biased, although they remain consistent. 53
  54. 54. Arthur CHARPENTIER - Sales forecasting. Estimation of parameters In classical linear econometrics, Y = Xβ + ε, with ε ∼ N(0, σ2 ). Then β = (X X)−1 X Y • is the ordinary least squares estimator, OLS, • is the maximum likelihood estimator, ML. Maximum likelihood estimator is consistent, asymptotically efficient, and (asymptotic) variances can be determined. This can be obtined using optimization techniques. Remark it is possible to use generalized method of moments, GMM. 54
  55. 55. Arthur CHARPENTIER - Sales forecasting. To go further, modeling a qualitative variable In some case, the variable of interest is not necessarily of price (continuous variable on R), but a binary variable. Consider the following regression model Yi = α + βXi + εi, with Yi =    1 0 where the ε are independent random variables, with 0 mean. Then E(Yi) = α + βXi. Note that Yi is then a Bernoulli (binomial) distribution. Classical models are either the probit or the logit model. The idea is that there exists a continuous latent unobservable Y ∗ such that Yi =    1 if Y∗ i > ti 0 if Y∗ i ≤ ti with Y ∗ i = α + βXi + εi, which is now a classical regression model. Equivalently, it means that Yi is then a Bernoulli (binomial) distribution B(pi) 55
  56. 56. Arthur CHARPENTIER - Sales forecasting. where pi = F(α + βXi), where F is a cumulative distribution function. If F is the cumulative distribution function of N(0,1), i.e. F(x) = 1 √ 2π x −∞ exp − z2 2 dz, which is the probit model, or the cumularive distribution of the logistic distribution F(x) = 1 1 + exp(−x) for the logit model. Those models can be extended to so-called ordered probit model, where Y can denote e.g. a rating (AAA,BB+, B-,...etc). Maximum likelihood techniques can be used. 56
  57. 57. Arthur CHARPENTIER - Sales forecasting. Modeling the random component The unpredictible random component is the key element when forecasting. Most of the uncertainty comes from this random component εt. The lower the variance, the smaller the uncertainty on forecasts. The general theoritical framework related to randomness of time series is related to weakly stationary. 57

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