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AT&T Revenue Forecast: Multiple Linear Regression
Model
August 7, 2016
Craig Jenkins
Introduction
The purpose, or proposal, of this project is to find variables that have a causal
relationship with AT&T Revenue in an attempt to create a reliable forecast for the next eight
quarters. The following function is the basis of what the relationship between the Y and X
variables is expected to be for the multiple linear regression model in the stated hypothesis.
AT&T Revenue = f (-Interest Rates, Verizon Revenue)
The function will be broken down by the reasoning to why the following X variables were
chosen as well as detailing the hypothesis statements in regards to the causal relationship
between each X variable and Y. The p-values between all correlations mentioned below is 0.000
which gives each independent variable more than the 95% confidence needed from an accuracy
and or risk standpoint.
AT&T Revenue (Y):
Before taking a look at the independent variables, there are some interesting
characteristics noticeable when looking at the time series plot of AT&T Revenue shown below.
From quarter 1-12, you see a fairly constant positive trend with little to no seasonality or cycle
involved. Quarters 12-14 has a prominent spike followed by constant positive trend with more
seasonality showing. Quarters 24-44 start out with a lot of seasonality and then becomes stable,
but the trend is now negative (eventually becoming flat). The main focus of this time series is
quarters 44-52 where there’s an exponentially positive jump in a relatively short amount of time.
The only logical reason for the dramatic increase would be the merger AT&T was involved in
with Cingular (occurred around the same time). It should be noted that the merger gap in the time
series mentioned may have an effect on the scatter plots that show a relationship between the
independent variables and Y shown later. However, correlation and other methods should help to
increase the accuracy of this time period going further. Lastly, in quarters 52-80 you can see a
more gradual positive trend with increasing seasonality as time progresses.
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AT&T
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Time Series Plot of AT&T Revenue
Interest Rates (X1):
From a logical standpoint, one can assume that interest rates as a whole is a macro
variable that tends to have an inverse relationship with most industries in the US economy. In
regards to the function, the belief is that interest rates will inversely impact the dependent
variable in either direction. The time series plot of interest rates shown below does indeed have a
negative trend and very high levels of cyclical involvement throughout. Seasonality is also
somewhat prevalent but the cycle is the dominant force.
The scatter plot showing the relationship between X1 and Y has a noticeable gap in the
middle which most likely can be explained by the merger mentioned previously. The correlation
line helps solve this problem by giving a better visualization of the trend between the two
variables which is strong and negative sloping as predicted.
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Inte
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Time Series Plot of Interest Rates
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Scatterplot of AT&T Revenue vs Interest Rates
The correlation matrix below shows that the relationship between X1 and Y is -0.825
which signifies a strong negative trending correlation between the two variables. The strong
correlation supports the assumption that as interest rates rise, revenue decreases and if interest
rates fall, revenue increases.
Correlation: AT&T Revenue, Interest Rat, Non Financial Corp Securities, Household Cr, Verizon Revenue
AT&T Revenue Interest Rates Non Financial Co Household CreditInterest Rates -0.825 0.000
Non Financial Co 0.729 -0.619 0.000 0.000
Household Credit 0.874 -0.828 0.691 0.000 0.000 0.000
Verizon Revenue 0.875 -0.844 0.781 0.890 0.000 0.000 0.000 0.000
Cell Contents: Pearson correlation P-Value
Verizon Revenue (X2):
The first three independent variables were macro-economic indicators/data so this
variable was added to give a look of how AT&T’s top competitor Verizon compares in terms of
sales revenue on a micro level. The time series for Verizon’s Revenue has a fairly constant
positive trend throughout the graph with exception to quarters 20-22 which show an exponential
increase and decrease in a short amount of time. The rapid increase followed by a rapid decrease
has to be explained by an activity not considered normal in standard operating procedures. From
a historical standpoint, the merger of Bell Atlantic with GTE to form Verizon Communications
took $52 billion dollars and two years to complete which fits the graph’s time frame (quarters
20-22, the year 2000). The biggest evidence of seasonality occurs between quarters 72-80 and
cycle does not seem to be present.
The scatter plot shows a strong positive trend amongst the compared variables Y and X2
which is to be expected since they are the two top companies (rivals) in the industry. There is
some grouping but, for the most part, the data still follows the positive correlation line (with
exception of one outlier data point).
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Time Series Plot of Verizon Revenue
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Scatterplot of AT&T Revenue vs Verizon Revenue
The correlations between X2 and other independent variables does not hold a lot of
significance since X2 is a variable that is expected to be closely similar to Y. That being said,
when looking at the correlation matrix, the correlation between Y and X4 is 0.875 which shows a
strong positive relationship between the two variables.
Correlation: AT&T Revenue, Interest Rat, Non Financial Corp Securities, Household Cr, Verizon Revenue
AT&T Revenue Interest Rates Non Financial Co Household CreditInterest Rates -0.825 0.000
Non Financial Co 0.729 -0.619 0.000 0.000
Household Credit 0.874 -0.828 0.691 0.000 0.000 0.000
Verizon Revenue 0.875 -0.844 0.781 0.890 0.000 0.000 0.000 0.000
Cell Contents: Pearson correlation P-Value
Overall, the cross correlations previously mentioned between X1 and X2 (to a lesser
extent) are of significance. The main focus in the model, however, are all the XY correlations
and scatter plots previously mentioned, which both support the alternate hypothesis statement
that:
Ho: AT&T Revenue ≠ f (- Interest Rates + Verizon Revenue)
Ha: AT&T Revenue = f (- Interest Rates + Verizon Revenue)
Methodology and Forecast Results
In this section, the reasoning behind why a certain univariate model was used for each X
variable will be explained (decomposition, exponential smoothing, ARIMA). The unique
dynamics of each x variable will then lead to explaining the causal impact and variation of
AT&T Revenue in the regression model.
Univariate Models (X Variables)
Interest Rates (X1):
For the X1 variable, the univariate model chosen to create the best forecasting model was
Winter’s Method Exponential Smoothing. The MAPE associated with the best model created
was lower than both the decomposition and ARIMA models of the X1 variable.
When looking at the time series plot of the X1 variable from a visual standpoint, there
seems to be evidence of cycle and trend with some seasonality. The auto correlation graph
demonstrates that seasonality is not a driving force with interest rates since the lines show a
relatively natural decrease every four quarters. However, every second quarter seems to have a
slightly increased seasonality while every fourth quarter seems to decrease in seasonality. With
seasonality being present, the winter’s method was used to obtain the best forecast possible.
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Inte
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Time Series Plot of Interest Rates
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Autocorrelation Function for Interest Rates(with 5% significance limits for the autocorrelations)
Taking a look at actual winter’s method for interest rates (X1) below, the Mean Absolute Percent
Error (MAPE) illustrates that there is a 7.54% chance for error in regards to the model. One can
also look at the Root Mean Square Error (RMSE) to repesent accuracy which calulates to .3750
points of error. I set the alpha weight to .9 to draw more so from the most recent data (timewise)
to have greater affect. Trend was set at .1 and seasonality was also set at .1 to pull back further
into the data to create a more accurate model.
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α (level) 0.9γ (trend) 0.1δ (seasonal) 0.1
Smoothing Constants
MAPE 7.54214MAD 0.2938MSD 0.14606
Accuracy Measures
Index
Inte
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Rate
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ActualFitsForecasts95.0% PI
Variable
Winters’ Method Plot for Interest RatesMultiplicative Method
When running autocorrelation and a histogram with a fitted line on the residuals of the model as
shown below, it is evident that the forecast is not significantly biased but leaves some seasonality
behind in the residuals.
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Autocorrelation Function for RESI1(with 5% significance limits for the autocorrelations)
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Histogram of RESI1Normal
The next step taken was to look at the LBQ numbers for the residuals pertaining to interest rates.
The 12th and 24th lag are below the significance levels which means that we can be more than
95% sure the model picked up Trend, Seasonality, and Cycle.
Autocorrelation Function: RESI1
Lag ACF T LBQ
12 0.005404 0.04 18.89 (less than 21 which shows no significance)
24 -0.044216 -0.31 32.22 (less than 36.4 which shows no significance)
Finally, from a visual standpoint, the forecast for 9 periods looks to fit very nicely with the rest
of the data and overall helps create a reliable forecasting model for the X1 variable. The final
time series plot with the forecasted data is as follows.
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Data
Interest RatesFORE1
Variable
Time Series Plot of Interest Rates, FORE1
Verizon Revenue (X2)
The univariate model that was used to create the best forecast for the X2 variable was the
ARIMA model. The ARIMA model had the lowest MAPE in comparison to the Decomposition
and Exponential smoothing models. The MA model of (0, 1, 1) (), 1, 1) gave the best results
more specifically.
From looking at both the time series plot and the ACF, there seems to be both seasonality
and trend. Therefore seasonality differencing will be implimented first.
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ueTime Series Plot of Verizon Revenue
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Autocorrelation Function for Verizon Revenue(with 5% significance limits for the autocorrelations)
The next step is too look at the PACF and the AFC of the seasonal difference that generated the
best results before giving to variation. The number of seasonal differences used to achieve such
data is 1. As seen below, both the PACF and the ACF were dying downtherefore the PDQ in
regards to the model will inititilly be (1, 1, 1).
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Autocorrelation Function for Verizon 1 seas diff(with 5% significance limits for the autocorrelations)
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Partial Autocorrelation Function for Verizon 1 seas diff(with 5% significance limits for the partial autocorrelations)
The second step is to difference trend completely out of the data to make it stationary. When
looking at the trend analysis graphs below, 1 difference seem to generate the best slope or the
data. Looking at the how the trend line flattens and centers near zero while also dropping the
slope value demonstrates this below.
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MAPE 2330MAD 1791MSD 14630527
Accuracy Measures
Index
Veriz
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Trend Analysis Plot for Verizon 1 seas diffLinear Trend ModelYt = 2121 - 14.7×t
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Ver 1
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Trend Analysis Plot for Ver 1 trend diffLinear Trend Model
Yt = -99 + 1.4×t
The next step is to look at the PACF and the ACFs to determine the p, q for the data and model
type. The PACF indicates indicates that it is dying down at a gradual rate which earns a value of
zero for the AR section. The ACF seems to have one major spike when looking at the first 3 lags
so the MA section will get a value of 1. Therefore. This model will be considered as an MA
model type with 1 difference and 1 coefficient (0, 1, 1).
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Partial Autocorrelation Function for Ver 1 trend diff(with 5% significance limits for the partial autocorrelations)
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Autocorrelation Function for Ver 1 trend diff(with 5% significance limits for the autocorrelations)
The final step is to look at the actual generated model for significance. It is worth noting that the
p, d, q was altered to (0, 1, 1 - MA model) because it generated more significance and better
accuracy. All the values involved in the model indicate that all coefficients are significant and
that it is a useful model to generate a gforecast. The constant was zero so it was not added in the
model (0, 1, 1) (0, 1, 1).
ARIMA Model: Verizon Revenue
Estimates at each iteration
Iteration SSE Parameters 0 1676566145 0.100 0.100 1 1318272564 0.247 0.250 2 1087603886 0.380 0.400 3 928715825 0.500 0.550 4 812302163 0.603 0.700 5 722172144 0.670 0.850 6 643502910 0.707 1.000 7 637840683 0.755 1.012 8 637296335 0.771 1.012 9 637204520 0.777 1.012 10 637186661 0.780 1.012 11 637183358 0.781 1.013 12 637182940 0.782 1.013
Relative change in each estimate less than 0.0010
Final Estimates of Parameters
Type Coef SE Coef T PMA 1 0.7820 0.0716 10.92 0.000 (p value below .05, t value above 1.96)SMA 4 1.0126 0.0362 28.00 0.000
Differencing: 1 regular, 1 seasonal of order 4Number of observations: Original series 82, after differencing 77Residuals: SS = 625414152 (backforecasts excluded) MS = 8338855 DF = 75
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48Chi-Square 2.0 4.2 6.2 8.6 (LBQ values less than 21, 36.4)DF 10 22 34 46P-Value 0.996 1.000 1.000 1.000 (P values less than .05)
Forecasts from period 82
95% LimitsPeriod Forecast Lower Upper Actual 83 33268.7 27607.6 38929.7 84 33630.6 27836.6 39424.5 85 33659.8 27735.9 39583.7 86 33976.6 27925.5 40027.7 87 34646.8 28485.0 40808.6 88 35008.7 28727.6 41289.8 89 35037.9 28639.7 41436.2 90 35354.7 28841.5 41867.9
The time series plot below is a combination of the original X2 data combined with the forecast
values generated from the forecasting model previously mentioned.
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Verizon Revenueforecast
Variable
Time Series Plot of Verizon Revenue, forecast
Regression Model Part 1 (Introduction)
The multiple linear regression model was used to define the linear relationship between
the Y variable (AT&T Revenue) and the X variables accordingly. The error derived from the X
variables were minimized by choosing the best univariate forecasting model to represent each
variable individually (as show above). The accuracy and acceptance of the regression model will
depend on the strength of the relationship between all variables and minimizing error.
Regression Model Part 2 (Seasonaility)
The first factor that needs to be checked for with regards to the Y variable is seasonality.
When looking at a time series plot, seasonality is present in certain segments but not all. When
looking at the ACF, it more accurately illustrates the fact that, overall seasonality is present but
not prevalent.
Regression Part 3 – Transformation
When trying to derive the best multiple regression model possible, transforming interest
rates by squaring it allowed for increased r2 and lowered the MAPE of the entire model (as
compared to previous attempts). The transformation was done in an attempt to increase cycle in
the model which will be elaborated on later. The scatter plot and and correlation matrix below
shows the results in the new relationship between the X and Y variable.
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Sq VR
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Scatterplot of AT&T Revenue vs Sq VR
Correlation: AT&T Revenue, Sq IR
Pearson correlation of AT&T Revenue and Sq IR = -0.799P-Value = 0.000
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Time Series Plot of AT&T Revenue
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Autocorrelation Function for AT&T Revenue(with 5% significance limits for the autocorrelations)
Regression Part 4 – Key Indicators (Dummy Variables)
Two specific dummy variables were used to enhance the accuracy and significance of the
regression model which I labeled as the following:
Business Model Change (BMC): The BMC variable accounted for the spike in AT&T Revenue
from periods 11-23. There was a significant enough jump in the data for it to be reasonable to
make a dummy variable as has been done. It is worth noting that the dummy variable was turned
on only during this time frame.
Merger: As been previously mentioned, in periods 43-49 there was a significant
merger/acquisition between AT&T and Singular which allowed for a significant jump in the data
during this time period. The dummy variable was turned on at the beginning of this time period
and was left on because of the impact that it has had on the rest of the data.
Regression Part 5 – Regression Model and Evaluation
When taking a look at the regression model below, a couple factors stick out. The p
values for all variables associated in the model were below .05 which gives us at least a 95 %
confidence of accuracy. All T values were also above 1.96 which indicates that all coefficients
are indeed significant. All the coefficients have positive correlations with the Y variable except
interest rates which is expected since they have an inverse relationship with each other. The r2
(adjusted) is 89.73% which indicates that the model below can account for 90% of the causal
relationship in respect to Y. The F value which tests the acceptability of the entire model is more
than three time the table value of 3.15 which equals 9.45 so the overall model is adequate.
F Actual Value 175.82 > Table Value 9.45
* F table Statistic (k=2; n-(2+1) = 79) = 3.15 * 3 = 9.45
Regression Analysis: AT&T Revenue versus Interest Rates, Sq VR, BMC, Merger
Method
Categorical predictor coding (1, 0)Rows unused 1
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-ValueRegression 4 8268974951 2067243738 175.82 0.000 Interest Rates 1 144582015 144582015 12.30 0.001 Sq VR 1 203795286 203795286 17.33 0.000 BMC 1 110725268 110725268 9.42 0.003 Merger 1 796779826 796779826 67.77 0.000Error 76 893606065 11757975Total 80 9162581016
Model Summary
S R-sq R-sq(adj) R-sq(pred)3428.99 90.25% 89.73% 85.19%
Coefficients
Term Coef SE Coef T-Value P-Value VIFConstant 15737 2961 5.31 0.000Interest Rates -1714 489 -3.51 0.001 3.45Sq VR 0.000011 0.000003 4.16 0.000 4.53BMC 1 3579 1166 3.07 0.003 1.26Merger 1 11075 1345 8.23 0.000 3.11
Regression Equation
BMC Merger0 0 AT&T Revenue = 15737 - 1714 Interest Rates + 0.000011 Sq VR
0 1 AT&T Revenue = 26812 - 1714 Interest Rates + 0.000011 Sq VR
1 0 AT&T Revenue = 19316 - 1714 Interest Rates + 0.000011 Sq VR
1 1 AT&T Revenue = 30392 - 1714 Interest Rates + 0.000011 Sq VR
Fits and Diagnostics for Unusual Observations
AT&TObs Revenue Fit Resid Std Resid 20 12906 20721 -7815 -3.68 R X 43 10304 23195 -12891 -3.91 R 44 12909 22506 -9597 -2.92 R 45 15756 23734 -7978 -2.39 R 46 15770 23174 -7404 -2.24 R 47 15638 23740 -8102 -2.44 R 48 15891 24267 -8376 -2.51 R
R Large residualX Unusual X
Durbin-Watson Statistic
Durbin-Watson Statistic = 0.652768
Given the results generated from the model, we can reject the null and accept the alternative
hypothesis of the multiple regression model.
Reject the Null; Ho: AT&T Revenue ≠ f (- Interest Rates + Verizon Revenue)
Accept Alternative; Ha: AT&T Revenue = f (- Interest Rates + Verizon Revenue)
Regression Part 6 - Model Investigation
When looking at the Durin-Watson statistic generated from the model, a number of .6527
indicates that positive serial correlation is occurring. The main concern with this phenomenon is
that some cycle, more than likely, has been left in the residuals. The X2 variable was thus squared
(previously mentioned) in an attempt to raise the DW statistic. Transforming the data did indeed
help raise the DW statistic and increased the r2. However, the DW range of 1.53 – 2.47 was not
achieved which will be an acceptable loss given the model’s strength overall. The ACF shows
that cycle is the only factor that has been left in the residuals.
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Autocorrelation Function for RESI(with 5% significance limits for the autocorrelations)
Heteroscedasticity isn’t noticeably present in the regression model when running a
regression model on the residuals (in relation to the fits). When looking at all of the significant
indicators (T value, P value, R2, F Test), there shows to be know relation to the fits of the model
and the residuals left behind which tells us that heteroscedasticity is not present in reference to
the regression model.
Regression Analysis: RESI versus FITS
Method
Rows unused 1
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-ValueRegression 1 0 0 0.00 1.000 FITS 1 0 0 0.00 1.000Error 79 893606065 11311469Total 80 893606065
Model Summary
S R-sq R-sq(adj) R-sq(pred)3363.25 0.00% 0.00% 0.00%
Coefficients
Term Coef SE Coef T-Value P-Value VIFConstant 0 794 0.00 1.000FITS -0.0000 0.0370 -0.00 1.000 1.00
Regression Equation
RESI = 0 - 0.0000 FITS
Fits and Diagnostics for Unusual Observations
Obs RESI Fit Resid Std Resid 20 -7815 0 -7815 -2.34 R 43 -12891 -0 -12891 -3.86 R 44 -9597 -0 -9597 -2.87 R 45 -7978 -0 -7978 -2.39 R 46 -7404 -0 -7404 -2.22 R 47 -8102 -0 -8102 -2.43 R 48 -8376 -0 -8376 -2.51 R
R Large residual
Durbin-Watson Statistic
Durbin-Watson Statistic = 0.652768
Multicollinearity can also be confidently ruled out of the regression model since the VIF
of all the variable coefficients are relatively low (all around 1 – 4). If one of the variables had a
VIF of 10 or above, there would be concern but there isn’t. It should also be worth noting that
seasonality is well represented in the model as a result of the VIF indicators.
Regression Part 7 RMSE and MAPE
RMSE = Square root (SUM (Residuals) 2) = 29893.2 point of error
MAPE = SUM (ABS (Residuals)/Actual Data) = 12.6032 % of error
Having a possibility of 12 % of error in reference to the MAPE is concerning but will be
accepted in the model. Given the tools available, one would believe that the reason for the
somewhat high MAPE is due to the inability to pick up more cycle from the residuals.
Transforming the X2 variable did help somewhat alleviate the problem but still left some cycle
behind.
Regression Part 8 - Remedies Model Inaccuracies
As mentioned in section 6 and 8, multicollinearity and heteroscedasticity are not
noticeably present in the model. Positive serial correlation however is the main area of concern
with the model and transforming the X2 variable to make cycle more prevalent was the solution
implemented. No other tools could be used without altering the effectiveness of the model (as
tried with previously failed model attempts).
Regression Part 9 – Evaluation of model Fit Residuals
When looking at the ACF and histogram with a fitted line, it’s evident that cycle is the
only business factor left in the residuals. Seasonality has been previously proven to not have
much presence in the residuals. The histogram shows some abnormality, however, in one section
which would be something to keep an eye on.
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Histogram of RESINormal
Regression Part 10 – Model Forecast
The following time series plot shows the regression forecast plus the original historical
data. The forecast numbers of the univariate models that helped create the regression forecast
will also be shown below. All of the forecasted data falls within the upper and lower confidence
limits.
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AT&T RevenueAT&T ForeAT&T LLAT&T UL
Variable
Time Series Plot of AT&T Revenue, AT&T Fore, AT&T LL, AT&T UL
Interest Rates Fcst Sq. Ver Rev Fcst BMC Fcst Merger Fcst
1.90983 1054275298 0 1
1.77329 1113130819 0 1
1.76009 1081607983 0 1
1.77452 1099790952 0 1
1.66705 1115374036 0 1
1.54047 1174229556 0 1
1.52116 1142706721 0 1
1.52517 1160889689 0 1
Conclusion
In regards to the regression model and the forecast it generated, the hold out of the
forecast is, indeed, acceptable. From a visual standpoint of the time series plot, the AT&T
Revenue forecast falls within the range given by the upper and lower confidence limits. The
confidence limits allow us to be 95 % confident in the results of the forecast. The RMSE and
MAPE leaves some concern to the model but overall when looking at the significance of the
model, it seems to be pretty affective.
From a business standpoint, the model indicates that a continual increase in revenue for
the company is expected for the next 8 periods. The trend of the forecasted data seems to be on
par with the trend experienced since the 63rd period. Seasonality will continue to play a role as
well when looking at the forecasted data. It should also be noted that before AT&T’s merger
with Singular, the slope was starting to falls at a somewhat concerning rate. I would advise in a
future time (after the forecasted period) that when the trend of A&T Revenue starts to flatten (or
decrease) again to start looking for another merger/ aquistion opportunity to simulate their
previous success.
An interesting factor in creating the multiple regression model is that without the 2
dummy variables added into the model (Merger and BMC), more macroeconomic variables held
significance in determining a causal relationship with AT&T Revenue. After implementing the
dummy variables, interest rates and Verizon Revenue were the only two variables (one being
micro and somewhat but not totally causal) that still held significance. From my conclusion, the
merger has become highly influential and causal, and in return, other macro economic variales
don’t have as much effect on the Y variable during this point in time. Therefore, the model will
need to be adjusted at some point in the future by turning the merger dummy variable off and
adding back in some more macroeconomic variables. The reasoning behind this is that at some
point, significant causation of the merger variable will shift importance back to the
macroeconomic variables.
Appendix
*All orignial data obtained from public sources (starting Q1 1995)
Exhibit A: (Verizon Revenue, and associated transformation, forecasts)
Verizon
Revenue
Verizon 1
seas diff
Ver 1 trend
diff
VR Fcst Sq VR Sq VR Fcst
3449.7 33268.68396 11900430.09 1054275298
3564.5 33630.57013 12705660.25 1113130819
3261.1 33659.81951 10634773.86 1081607983
3154.2 33976.614 9948977.64 1099790952
7044.0 3594.3 34646.80181 49617936 1115374036
7329.4 3764.8999 170.5999 35008.68798 53720102.89 1174229556
7376.6 4115.5 350.6001 35037.93736 54414229.04 1142706721
7405.2 4251.0002 135.5002 35354.73185 54836990 1160889689
7416.5 372.5 -3878.5002 55004472.25
7707.8 378.3999 5.8999 59410177.76
7373.9 -2.7002 -381.1001 54374399.74
7695.7 290.5 293.2002 59223801.57
7651.1 234.6001 -55.8999 58539332.74
7927.8 220 -14.6001 62850009.67
7909.9 536 316 62566516.43
8077.1 381.4009 -154.5991 65239562.18
7967.0 315.8999 -65.501 63473089
8295.0 367.2002 51.3003 68807025
8304.0 394.1001 26.8999 68956416
33628.0 25550.8989 25156.7988 1130842384
14532.0 6565 -18985.8989 211179024
16769.0 8474 1909 281199361
16533.0 8229 -245 273340089
16873.0 -16755 -24984 284698129
16266.0 1734 18489 264582756
16909.0 140 -1594 285914281
17004.0 471 331 289136016
17011.0 138 -333 289374121
16430.0 164 26 269944900
16752.0 -157 -321 280629504
17113.0 109 266 292854769
17009.0 -2 -111 289306081
16490.0 60 62 271920100
16829.0 77 17 283215241
17063.0 -50 -127 291145969
17278.0 269 319 298529284
17056.0 566 297 290907136
17758.0 929 363 315346564
18206.0 1143 214 331458436
18263.0 985 -158 333537169
18179.0 1123 138 330476041
18053.0 295 -828 325910809
18486.0 280 -15 341732196
17927.0 -336 -616 321377329
21231.0 3052 3388 450755361
21886.0 3833 781 478996996
22459.0 3973 140 504406681
22606.0 4679 706 511031236
22584.0 1353 -3326 510037056
23273.0 1387 34 541632529
23772.0 1313 -74 565107984
23840.0 1234 -79 568345600
23833.0 1249 15 568011889
24124.0 851 -398 581967376
24752.0 980 129 612661504
24645.0 805 -175 607376025
26591.0 2758 1953 707081281
26861.0 2737 -21 721513321
27265.0 2513 -224 743380225
27091.0 2446 -67 733922281
26913.0 322 -2124 724309569
26773.0 -88 -410 716793529
26484.0 -781 -693 701402256
26395.0 -696 85 696696025
26990.0 77 773 728460100
27536.0 763 686 758231296
27913.0 1429 666 779135569
28436.0 2041 612 808606096
28242.0 1252 -789 797610564
28552.0 1016 -236 815216704
29007.0 1094 78 841406049
30045.0 1609 515 902702025
29420.0 1178 -431 865536400
29786.0 1234 56 887205796
30279.0 1272 38 916817841
31065.0 1020 -252 965034225
30818.0 1398 378 949749124
31483.0 1697 299 991179289
31586.0 1307 -390 997675396
33192.0 2127 820 1101708864
31984.0 1166 -961 1022976256
32224.0 741 -425 1038386176
Exhibit B: (Interest Rates, Forecasts)
IR IR Fcst
7.483 1.667050163
6.620 1.540470384
6.323 1.521159896
5.893 1.525169962
5.910 1.424270251
6.720 1.307648112
6.780 1.282228296
6.343 1.275816899
6.563
6.697
6.243
5.907
5.587
5.597
5.203
4.670
4.983
5.540
5.883
6.140
6.480
6.177
5.893
5.567
5.050
5.270
4.980
4.770
5.077
5.100
4.260
4.007
3.920
3.620
4.233
4.287
4.020
4.600
4.303
4.173
4.297
4.160
4.213
4.490
4.570
5.070
4.897
4.630
4.680
4.847
4.730
4.260
3.663
3.887
3.863
3.253
2.737
3.313
3.517
3.460
3.717
3.490
2.787
2.863
3.460
3.210
2.427
2.047
2.037
1.823
1.643
1.707
1.950
1.997
2.710
2.747
2.763
2.623
2.497
2.280
1.967
Exhibit C: (Regression MAPE, RMSE, Dummy Variables, Forecasts)
BMC BMC
Fcst
Merger Merger
Fcst
FITS RESI MAPE RMSE
0 0 0 1 3034.69
5
2129.305 12.6031
5
247.2297
0 0 0 1 4523.11
9
732.8808
0 0 0 1 5009.82 557.1801
0 0 0 1 5739.69
3
-14.6931
0 0 0 1 6129.60
6
-555.606
0 0 0 1 4784.37
6
953.6241
0 0 0 1 4688.84
6
1268.154
0 0 0 1 5441.84
2
775.1579
0 0 5066.48
4
906.5161
0 0 4884.40
1
1036.599
1 0 9187.52
7
-2858.53
1 0 9815.8 -1357.8
1 0 10357.1
3
680.8749
1 0 10385.4 1012.542
6
1 0 11056.7
2
549.2789
1 0 11999.1
6
165.8375
1 0 11443.4
1
375.5901
1 0 10545.4
4
1722.561
1 0 9958.47
2
2586.528
1 0 20720.7
4
-7814.74
1 0 10436.0
2
2116.976
1 0 11694.6
7
1496.329
1 0 12097.4
5
1324.548
0 0 9198.10
2
3040.898
0 0 9871.56
9
1318.431
0 0 9719.47
8
1757.522
0 0 10250.5
9
1087.415
0 0 10613.0
8
1289.92
0 0 9882.42
4
639.5758
0 0 9955.14
2
887.8579
0 0 11524.0
4
-968.043
0 0 11920.8
7
-703.871
0 0 11886.0
2
-1553.02
0 0 12519.4
4
-2315.44
0 0 11551.7
3
-1312.73
0 0 11538.1
9
-1471.19
0 0 11914.9 -1786.91
1
0 0 11178.4
9
-982.488
0 0 11857.0
1
-1565.01
0 0 12101.7
8
-1814.78
0 0 11858.0
7
-1610.07
0 0 12044.1
8
-1727.18
0 1 23194.9
3
-12890.9
0 1 22505.9
3
-9596.93
0 1 23733.6
5
-7977.65
0 1 23174.4
8
-7404.48
0 1 23739.6
7
-8101.67
0 1 24266.6
7
-8375.67
0 1 24170.4
8
4798.523
0 1 24218.0
9
5259.911
0 1 24665.7
3
5466.269
0 1 25505.5
6
4843.438
0 1 26524.8
5
4219.15
0 1 26289.2
3
4576.767
0 1 26653.0
4
4688.965
0 1 27642.9
4
3433.059
0 1 29580.4
4
990.5571
0 1 28744.1
7
1869.832
0 1 28626.3 2107.705
0 1 28623.6
6
2084.342
0 1 28082.2
7
2447.73
0 1 28391.5
3
2416.466
0 1 29434.8
2
2146.178
0 1 29253.7
5
2107.248
0 1 28566.0
4
2680.965
0 1 29308.6
5
2186.346
0 1 30871.9
7
606.0252
0 1 31834.2
7
668.7315
0 1 31735.4
1
86.58556
0 1 32286.8
5
-711.845
0 1 32871.6
8
-1412.68
0 1 33409.7 -831.752
5
0 1 32600.5
5
-1244.55
0 1 32749.1
6
-674.157
0 1 31838.7
5
319.2529
0 1 32284.5
5
878.4538
0 1 32094.7
3
381.2723
0 1 32771.7
8
-196.779
0 1 33057.4
4
-100.442
0 1 34526.3
4
-87.3419
0 1 34232.8
8
-1656.88
0 1
Exhibit D: (AT&T Revenue, Upper and Lower Limit, Forecast)
AT&T AT&T Rev Lower Limit Upper Limit
Revenue Fcst
5164.0 35076.66817 33261.99651 36891.33984
5256.0 35914.54143 33931.08457 37897.99828
5567.0 35615.09697 33700.02963 37530.16431
5725.0 35800.04206 33845.21878 37754.86534
5574.0 36137.39677 34119.11269 38155.68085
5738.0 36958.20059 34763.75636 39152.64483
5957.0 36669.22878 34547.12901 38791.32855
6217.0 36872.03839 34704.25003 39039.82675
5973.0
5921.0
6329.0
8458.0
11038.0
11398.0
11606.0
12165.0
11819.0
12268.0
12545.0
12906.0
12553.0
13191.0
13422.0
12239.0
11190.0
11477.0
11338.0
11903.0
10522.0
10843.0
10556.0
11217.0
10333.0
10204.0
10239.0
10067.0
10128.0
10196.0
10292.0
10287.0
10248.0
10317.0
10304.0
12909.0
15756.0
15770.0
15638.0
15891.0
28969.0
29478.0
30132.0
30349.0
30744.0
30866.0
31342.0
31076.0
30571.0
30614.0
30734.0
30708.0
30530.0
30808.0
31581.0
31361.0
31247.0
31495.0
31478.0
32503.0
31822.0
31575.0
31459.0
32578.0
31356.0
32075.0
32158.0
33163.0
32476.0
32575.0
32957.0
34439.0
32576.0
33015.0
