A Study On Business Forecasting Statistics Essay
The aim of this report is to show my understanding of business forecasting using data which was drawn from the UK national statistics. It is a quarterly series of total consumer credit gross lending in the UK from the second quarter 1993 to the second quarter 2009.
The report answers four key questions that are relevant to the coursework.
In this section the data will be examined, looking for seasonal effects, trends and cycles. Each time period represents a single piece of data, which must be split into trend-cycle and seasonal effect. The line graph in Figure 1 identifies a clear upward trend-cycle, which must be removed so that the seasonal effect can be predicted.
Figure 1 displays long-term credit lending in the UK, which has recently been hit by an economic crisis. Figure 2 also proves there is evidence of a trend because the ACF values do not come down to zero. Even though the trend is clear in Figure 1 and 2 the seasonal pattern is not. Therefore, it is important the trend-cycle is removed so the seasonal effect can be estimated clearly. Using a process called differencing will remove the trend whilst keeping the pattern.
Drawing scattering plots and calculating correlation coefficients on the differenced data will reveal the pattern repeat.
Scatter Plot correlation
The following diagram (Figure 3) represents the correlation between the original credit lending data and four lags (quarters). A strong correlation is represented by is showed by a straight-line relationship.
As depicted in Figure 3, the scatter plot diagrams show that the credit lending data against lag 4 represents the best straight line. Even though the last diagram represents the straightest line, the seasonal pattern is still unclear. Therefore differencing must be used to resolve this issue.
Differencing
Differencing is used to remove a trend-cycle component. Figure 4 results display an ACF graph, which indicates a four-point pattern repeat. Moreover, figure 5 shows a line graph of the first difference. The graph displays a four-point repeat but the trend is still clearly apparent. To remove the trend completely the data must differenced a second time.
First differencing is a useful tool for removing non-stationary. However, first differencing does not always eliminate non-stationary and the data may have to be differenced a second time. In practice, it is not essential to go beyond second differencing, because real data generally involve non-stationary of only the first or second level.
Figure 6 and 7 displays the second difference data. Figure 6 displays an ACF graph of the second difference, which reinforces the idea of a four-point repeat. Suffice to say, figure 7 proves the trend-cycle component has been completely removed and that there is in fact a four-point pattern repeat.
Question 2
Multiple regression involves fitting a linear expression by minimising the sum of squared deviations between the sample data and the fitted model. There are several models that regression can fit. Multiple regression can be implemented using linear and nonlinear regression. The following section explains multiple regression using dummy variables.
Dummy variables are used in a multiple regression to fit trends and pattern repeats in a holistic way. As the credit lending data is now seasonal, a common method used to handle the seasonality in a regression framework is to use dummy variables. The following section will include dummy variables to indicate the quarters, which will be used to indicate if there are any quarterly influences on sales. The three new variables can be defined:
Q1 = first quarter
Q2 = second quarter
Q3 = third quarter
Trend and seasonal models using model variables
The following equations are used by SPSS to create different outputs. Each model is judged in terms of its adjusted R2.
Linear trend + seasonal model
Data = a + c time + b1 x Q1 + b2 x Q2 + b3 x Q3 + error
Quadratic trend + seasonal model
Data = a + c time + b1 x Q1 + b2 x Q2 + b3 x Q3 + error
Cubic trend + seasonal model
Data = a + c time + b1 x Q1 + b2 x Q2 + b3 x Q3 + error
Initially, data and time columns were inputted that displayed the trends. Moreover, the sales data was regressed against time and the dummy variables. Due to multi-collinearity (i.e. at least one of the variables being completely determined by the others) there was no need for all four variables, just Q1, Q2 and Q3.
Linear regression
Linear regression is used to define a line that comes closest to the original credit lending data. Moreover, linear regression finds values for the slope and intercept that find the line that minimizes the sum of the square of the vertical distances between the points and the lines.
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
.971a
.943
.939
3236.90933
Figure 8. SPSS output displaying the adjusted coefficient of determination R squared
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
B
Std. Error
Beta
1
(Constant)
17115.816
1149.166
14.894
.000
time
767.068
26.084
.972
29.408
.000
Q1
-1627.354
1223.715
-.054
-1.330
.189
Q2
-838.519
1202.873
-.028
-.697
.489
Q3
163.782
1223.715
.005
.134
.894
Figure 9
The adjusted coefficient of determination R squared is 0.939, which is an excellent fit (Figure 8). The coefficient of variable ‘time’, 767.068, is positive, indicating an upward trend. All the coefficients are not significant at the 5% level (0.05). Hence, variables must be removed. Initially, Q3 is removed because it is the least significant variable (Figure 9). Once Q3 is removed it is still apparent Q2 is the least significant value. Although Q3 and Q2 is removed, Q1 is still not significant. All the quarterly variables must be removed, therefore, leaving time as the only variable, which is significant.
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
B
Std. Error
Beta
1
(Constant)
16582.815
866.879
19.129
.000
time
765.443
26.000
.970
29.440
.000
Figure 10
The following table (Table 1) analyses the original forecast against the holdback data using data in Figure 10. The following equation is used to calculate the predicted values.
Predictedvalues = 16582.815+765.443*time
Original Data
Predicted Values
50878.00
60978.51
52199.00
61743.95
50261.00
62509.40
49615.00
63274.84
47995.00
64040.28
45273.00
64805.72
42836.00
65571.17
43321.00
66336.61
Table 1
Suffice to say, this model is ineffective at predicting future values. As the original holdback data decreases for each quarter, the predicted values increase during time, showing no significant correlation.
Non-Linear regression
Non-linear regression aims to find a relationship between a response variable and one or more explanatory variables in a non-linear fashion.
(Quadratic)
Model Summaryb
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
.986a
.972
.969
2305.35222
Figure 11
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
B
Std. Error
Beta
1
(Constant)
11840.996
1099.980
10.765
.000
time
1293.642
75.681
1.639
17.093
.000
time2
-9.079
1.265
-.688
-7.177
.000
Q1
-1618.275
871.540
-.054
-1.857
.069
Q2
-487.470
858.091
-.017
-.568
.572
Q3
172.861
871.540
.006
.198
.844
Figure 12
The quadratic non-linear adjusted coefficient of determination R squared is 0.972 (Figure 11), which is a slight improvement on the linear coefficient (Figure 8). The coefficient of variable ‘time’, 1293.642, is positive, indicating an upward trend, whereas, ‘time2?, is -9.079, which is negative. Overall, the positive and negative values indicate a curve in the trend.
All the coefficients are not significant at the 5% level. Hence, variables must also be removed. Initially, Q3 is removed because it is the least significant variable (Figure 9). Once Q3 is removed it is still apparent Q2 is the least significant value. Once Q2 and Q3 have been removed it is obvious Q1 is under the 5% level, meaning it is significant (Figure 13).
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
B
Std. Error
Beta
1
(Constant)
11698.512
946.957
12.354
.000
time
1297.080
74.568
1.643
17.395
.000
time2
-9.143
1.246
-.693
-7.338
.000
Q1
-1504.980
700.832
-.050
-2.147
.036
Figure 13
Table 2 displays analysis of the original forecast against the holdback data using data in Figure 13. The following equation is used to calculate the predicted values:
QuadPredictedvalues = 11698.512+1297.080*time+(-9.143)*time2+(-1504.980)*Q1
Original Data
Predicted Values
50878.00
56172.10
52199.00
56399.45
50261.00
55103.53
49615.00
56799.29
47995.00
56971.78
45273.00
57125.98
42836.00
55756.92
43321.00
57379.54
Table 2
Compared to Table 1, Table 2 presents predicted data values that are closer in range, but are not accurate enough.
Non-Linear model (Cubic)
Model Summaryb
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
.997a
.993
.992
1151.70013
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
B
Std. Error
Beta
1
(Constant)
17430.277
710.197
24.543
.000
time
186.531
96.802
.236
1.927
.060
time2
38.217
3.859
2.897
9.903
.000
time3
-.544
.044
-2.257
-12.424
.000
Q1
-1458.158
435.592
-.048
-3.348
.002
Q2
-487.470
428.682
-.017
-1.137
.261
Q3
12.745
435.592
.000
.029
.977
Figure 15
The adjusted coefficient of determination R squared is 0.992, which is the best fit (Figure 14). The coefficient of variable ‘time’, 186.531, and ‘time2?, 38.217, is positive, indicating an upward trend. The coefficient of ‘time3? is -.544, which indicates a curve in trend. All the coefficients are not significant at the 5% level. Hence, variables must be removed. Initially, Q3 is removed because it is the least significant variable (Figure 15). Once Q3 is removed it is still apparent Q2 is the least significant value. Once Q3 and Q2 have been removed Q1 is now significant but the ‘time’ variable is not so it must also be removed.
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
B
Std. Error
Beta
1
(Constant)
18354.735
327.059
56.120
.000
time2
45.502
.956
3.449
47.572
.000
time3
-.623
.017
-2.586
-35.661
.000
Q1
-1253.682
362.939
-.042
-3.454
.001
Figure 16
Table 3 displays analysis of the original forecast against the holdback data using data in Figure 16. The following equation is used to calculate the predicted values:
CubPredictedvalues = 18354.735+45.502*time2+(-.623)*time3+(-1253.682)*Q1
Original Data
Predicted Values
50878.00
49868.69
52199.00
48796.08
50261.00
46340.25
49615.00
46258.51
47995.00
44786.08
45273.00
43172.89
42836.00
40161.53
43321.00
39509.31
Table 3
Suffice to say, the cubic model displays the most accurate predicted values compared to the linear and quadratic models. Table 3 shows that the original data and predicted values gradually decrease.
Question 3
Box Jenkins is used to find a suitable formula so that the residuals are as small as possible and exhibit no pattern. The model is built only involving a few steps, which may be repeated as necessary, resulting with a specific formula that replicates the patterns in the series as closely as possible and also produces accurate forecasts.
The following section will show a combination of decomposition and Box-Jenkins ARIMA approaches.
For each of the original variables analysed by the procedure, the Seasonal Decomposition procedure creates four new variables for the modelling data:
SAF: Seasonal factors
SAS: Seasonally adjusted series, i.e. de-seasonalised data, representing the original series with seasonal variations removed.
STC: Smoothed trend-cycle component, which is smoothed version of the seasonally adjusted series that shows both trend and cyclic components.
ERR: The residual component of the series for a particular observation
Autoregressive (AR) models can be effectively coupled with moving average (MA) models to form a general and useful class of time series models called autoregressive moving average (ARMA) models,. However, they can only be used when the data is stationary. This class of models can be extended to non-stationary series by allowing differencing of the data series. These are called autoregressive integrated moving average (ARIMA) models.
The variable SAS will be used in the ARIMA models because the original credit lending data is de-seasonalised. As the data in Figure 19 is de-seasonalised it is important the trend is removed, which results in seasonalised data. Therefore, as mentioned before, the data must be differenced to remove the trend and create a stationary model.
Model Statistics
Model
Number of Predictors
Model Fit statistics
Ljung-Box Q(18)
Number of Outliers
Stationary R-squared
Normalized BIC
Statistics
DF
Sig.
Seasonal adjusted series for creditlending from SEASON, MOD_2, MUL EQU 4-Model_1
0
.485
14.040
18.693
15
.228
0
Model Statistics
Model
Number of Predictors
Model Fit statistics
Ljung-Box Q(18)
Number of Outliers
Stationary R-squared
Normalized BIC
Statistics
DF
Sig.
Seasonal adjusted series for creditlending from SEASON, MOD_2, MUL EQU 4-Model_1
0
.476
13.872
16.572
17
.484
0
ARMA (3,2,0)
Original Data
Predicted Values
50878.00
50335.29843
52199.00
50252.00595
50261.00
50310.44277
49615.00
49629.75233
47995.00
