Sales forecasting # 1
Arthur Charpentier
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
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”.
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.
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).
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.
−4 −2 0 2 4 6 8 American Express University of North Carolina Goldman Sachs PNC Financial Kudlow & co
Figure 1: Economic growth forecasts, from Wall Street Journal, Sept. 12, 2002,
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.
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
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●●●●●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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Daily log return
Level of the Nasdaq index
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,
The secular trends are long-run trends that cause changes in an economic data series,
three different patterns can be distinguished, • linear trend, bYt = α + βt
• constant rate of growth trend, bYt = Y0(1 + γ)t
• declining rate of growth trend, bYt = 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 bYt = log Y0 + log(1 + γ) · t, which is a linear model on the logarithm of the
Series decomposition
For those two models, standard regression techniques can be used.
For declining rate of growth trend, log bYt = 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,
b
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.
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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
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 bYi and Yi, which should be as closed as
possible. E.g. minimize
n
X
i=1
(Yi − bYi)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
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.
RM SE = v u u t 1 n n X i=1 (Yi − bYi)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.
● ● ESTIMATION PERIOD EX−POST FORECAST PERIOD EX−ANTE FORECAST PERIOD
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
Lest square estimation
Parameters are estimated using ordinary least squares techniques, i.e. b β = (X0X)−1X0Y . E( bβ) = β. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 10 15 20 25 0 20 40 60 80 100 120 car speed distance
Linear regression, distance versus speed
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 10 15 20 25 0 20 40 60 80 100 120 car speed distance
Linear regression, speed versus distance
Lest square estimation
Assuming ε ∼ N (0, σ2), then V ( bβ) = (X0X)−1σ2.
The variance of residuals σ2 can be estimated using εb0ε/(n − k − 1).b It is possible to test H0 : βi = 0, then bβi/σ
q
(X0X)−1i,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
b βi − tn−k(1 − α/2)σb q [(X0X)−1]i,i; bβi + tn−k(1 − α/2)bσ q [(X0X)−1]i,i
where tn−k(1 − α/2) stands for the (1 − α/2) quantile of the t distribution with
Lest square estimation
3.5 4.0 4.5 5.0 5.5 −0.01 0.01 0.02 0.03 0.04 Endemics Area ● −0.15 −0.10 −0.05 0.00 0.05 −0.01 0.01 0.02 0.03 0.04 Elevation Area ●Lest square estimation
The R2 is the correlation coefficient between series {Y1, · · · , Yn} and { bY1, · · · , bYn},
where bYi = Xiβ. It can be interpreted as the ratio of the variance explained byb regression, and total variance.
The adjusted R2, called R2, is defined as R2 = (n − 1)R 2 − k n − k = 1 − n − 1 n − k − 1(1 − R 2 ).
Assume that residuals are N (0, σ2), then Y ∼ N (Xβ, σ2I), and thus, it is possible to use maximum likelihood technique,
log L(β, σ|X, Y ) = −n 2 log(2π) − n 2 log(σ 2 ) − (Y − Xβ) 0(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
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 sk 2 + n 24[κ − 3] 2 ,
where sk denotes the empirical skewness, and κ the empirical kurtosis. Under assumption H0 of normality, BJ ∼2 (2).
Residual in linear regression
1 2 3 4 5 6 −3 −2 −1 0 1 2 3 Fitted values Residuals ● ● ● ● ● ● ● ● ● ● ● ● ● ● lm(Y ~ X1 + X2) Residuals vs Fitted 2 5 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1 0 1 −1 0 1 2 Theoretical Quantiles Standardized residuals lm(Y ~ X1 + X2) Normal Q−Q 2 5 1Since the future observation should be x00βb+ε (where ε is unknown, but yield additional uncertainty), the confidence interval for this predicted value can be computed as b βi − tn−k(1 − α/2)σb q 1+x00(X0X)−1x0; bβi + tn−k(1 − α/2)σb q 1+x00(X0X)−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
b βi − tn−k(1 − α/2)σb q x00(X0X)−1x0; bβi + tn−k(1 − α/2)bσ q x00(X0X)−1x0
Prediction in the linear model ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 10 15 20 25 0 20 40 60 80 100 120 car speed distance
Confidence and prediction bands
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 10 15 20 25 0 20 40 60 80 100 120 car speed distance
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 =b 1 n
n
X
i=1
Xi, and the 95% confidence interval is
b µ + u2.5% √σ n;µ + ub 97.5% σ √ n
where u2.5% = −1.9645 and u97.5% = 1.9645. Both are quantiles of the N (0, 1) distribution.
Regression, basics on statistical regression techniques
Step 2: in case σ is unknown. The natural estimate of unkown µ is still b µ = 1 n n X i=1
Xi, and the 95% confidence interval is
b µ + t2.5% √σb n;µ + tb 97.5% b σ √ n
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.
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
b
YT +1 = E(YT +1) = α + βXT +1
This yields a forecast error, εbT +1 = bYT +1 − YT +1. This error has two properties
• the forecast should be unbiased E(εbT +1) = 0
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 bYT +1 = α + bb βXT +1
Thus, the forecast error is then b
εT +1 = bYT +1 − YT +1 = (α − α) + ( bb β − β)XT +1 − εT +1
Thus, there are two sources of error: • the additive error term εT +1
Figure 14: Forecasting techniques, problem of uncertainty related to parameter estimation.
Goal of ordinay least squares, minimize PNI=1(Yi − bYi)2 where bY = α + βX. Then b β = nP XiYi − P Xi P Yi nP Xi2 − (P Xi) 2 and b α = P Yi n − bβ · P Xi n = Y − bβX The least square slope can be writen
b
β = P(Xi − X)(Yi − Y ) P(Xi − X)2
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,
minimum variance, of all linear unbiased estimators (i.e. BLUE, best linear unbiased estimators).
The two estimators are further asymptotically normal, √ n( bβ − β)→N 0, n · σ 2 P(Xi − X)2 and √n(α − α)→Nb 0, σ2 P X 2 i P(Xi − X)2 .
The asymptotic variances of α and bb β can be estimated as b V ( bβ) = σb 2 P(Xi − X)2 and bV (α) =b b σ2 nP(Xi − X)2
while the covariance is
c
cov(α, bb β) = −Xσb
2
Regression model and Gauss-Markov theorem
Thus, if σ denotes the standard deviation of εT +1, the standard deviation s of
b εT +1 can be estimated as b s2 = σb 1+ 1 T + (XT +1 − X)2 P(Xi − X)2 > σ.b
RM SE = u u t 1 n n X i=1 (Yi − bYi)2
Another useful statistic is Theil inequality coefficient defined as
U = v u u t 1 T n X i=1 (Yi − bYi)2 v u u t 1 T n X i=1 b Yi2 + v u u t 1 T n X i=1 Yi2
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.
Step 3, assume that α, β and XT +1 are unknown, but that b
XT +1 = XT +1 + uT +1, where uT +1 ∼ N (0, σu2). The two errors are uncorrelated.
Here, the error of forecast is b
εT +1 = bYT +1 − YT +1 = (α − α) + ( bb β − β)XT +1 − εT +1
It can be proved (easily) that E(εbT +1) = 0. But its variance is slightly more
complecated to derive
V (εbT +1) = V (α) + 2Xb T +1cov(α, bb β) + (XT +12 +σu2)V ( bβ) + σ2+β2σu2
And therefore, the forecast error variance is then s2 = σ 1 + 1 T + (XT +1 − X)2 + σu2 P(Xi − X)2 + β 2 σu2 > σb2, which,again, increases the forecast error.
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,
• 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
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.
b
YT +1 = α + βXT +1 + εbT +1 = α + βXT +1 + ρεT
assuming that εbT +1 = ρεT. Recursively,
b
εT +3 = ρεbT +2 = ρ3εT
b
εT +h = ρεbT +h−1 = ρhεT
Since |ρ| < 1, ρh approaches 0 as h gets arbitrary large. Hence, the information provided by serial correlation becomes less and less usefull.
b
YT +1 = α(1 − ρ) + βXT +1 + ρ(YT − βXT)
Since YT = α + βXT + εT, then
b
YT +1 = α + βXT +1 + ρεT
Thus, the forecast error is then b
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.
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 + ω2Xt−2 + ω3Xt−3 + · · · + ωhXt−h + · · · + εt, with 0 < ω < 1. Note that Yt−1 = α + β Xt−1 + ωXt−2 + ω2Xt−3 + ω3Xt−4 + · · · + ωhXt−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.
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
Estimation of parameters
In classical linear econometrics, Y = Xβ + ε, with ε ∼ N (0, σ2). Then b
β = (X0X)−1X0Y
• 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.
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 Yi∗ = α + βXi + εi, which is now a classical
regression model.
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 Z x −∞ exp −z 2 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).
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.
