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# IBM401 Lecture 12

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### Transcript

• 1. Quantitative Analysis for Business
Lecture 12 – Final Exam Review
October 4th, 2010
http://www.slideshare.net/saark/ibm401-lecture-12
• 2. Final exam
No. of questions:
3
Duration:
3 hours
Topics:
Everything
• 3. Linear regression
Identifying independent and dependent variable
The sample correlation coefficient for two variables
Linear regression model form
• 4. Linear regression
The assumptions of linear regression model are the following:
A linear relations exists between the dependent variable and the independent variable.
The independent variable is not random.
The expected value of the error term is 0.
The variance of the error term is the same for all observations
The error term is uncorrelated across observations.
The error term is normally distributed.
• 5. Linear regression
In the regression model Yi = bo + b1Xi + Ei, if we know the estimated parameters, bo and b1, for any value of the independent variable, X, then the predicted value of the dependent variable Y is Y = b0 + b1X
In simple linear regression, the F-test tests the same null hypothesis as testing the statistical significance of b1 (or the t-test)
H0: b1 = 0
H1: b1 ≠ 0
If F > Fc, then reject H0
^
^
• 6. Multiple regression
Multiple regression model
Intercept – the value of the dependent variable when the independent variables are all equal to zero
Each slope coefficient – the estimated change in the dependent variable for a one-unit change in the independent variable, holding the other independent variables contant
Y = 0 + 1X1 + 2X2 + … + kXk + 
• 7. Hypothesis testing of regression coefficients
t-statistic – used to test the significance of the individual coefficient in a multiple regression
t-statistic has n-k-1 degrees of freedom
Estimated regression coefficient – hypothesized value
Coefficient standard error of bj
• 8. F-statistic
F-test assesses how well the set of independent variables, as a group, explains the variation of the dependent variable
F-statistic is used to test whether at least one of the independent variables explains a significant portion of the variation of the dependent variable
• 9. Coefficient of determination (R2)
Multiple coefficient of determination, R2, can be used to test the overall effectiveness of the entire set of independent variables in explaining the dependent variable.
Unfortunately, R2 by itself may not be a reliable measure of the multiple regression model
R2 almost always increases as variables are added to the model
We need to take new variables into account
Where
n = number of observations
k = number of independent variables
Whenever there is more than 1 independent variable
Ra2 is less than or equal to R2
So adding new variables to the model will increase R2 but may increase or decrease the Ra2
Ra2 maybe less than 0 if R2 is low enough
• 12. Time-Series Models
Time-series models attempt to predict the future based on the past
Common time-series models are
Moving average
Exponential smoothing
Trend projections
Decomposition
Regression analysis is used in trend projections and one type of decomposition model
• 13. Decomposition of a Time-Series
A time series typically has four components
Trend (T) is the gradual upward or downward movement of the data over time
Seasonality (S) is a pattern of demand fluctuations above or below trend line that repeats at regular intervals
Cycles (C) are patterns in annual data that occur every several years
Random variations (R) are “blips” in the data caused by chance and unusual situations
• 14.
• Mathematically
where
wi = weight for the ith observation
Weighted Moving Averages
• Weighted moving averages use weights to put more emphasis on recent periods
• 15. Often used when a trend or other pattern is emerging
• Exponential Smoothing
Exponential smoothing is easy to use and requires little record keeping of data
It is a type of moving average
New forecast = Last period’s forecast
+ (Last period’s actual demand - Last period’s forecast)
Where  is a weight (or smoothing constant) with a value between 0 and 1 inclusive
• 16. Exponential Smoothing
Mathematically
where
Ft+1 = new forecast (for time period t + 1)
Ft = previous forecast (for time period t)
 = smoothing constant (0 ≤  ≤ 1)
Yt = pervious period’s actual demand
• The idea is simple – the new estimate is the old estimate plus some fraction of the error in the last period
• Exponential Smoothing with Trend Adjustment
Like all averaging techniques, exponential smoothing does not respond to trends
A more complex model can be used that adjusts for trends
The basic approach is to develop an exponential smoothing forecast then adjust it for the trend
Forecast including trend (FITt) = New forecast (Ft)
+ Trend correction (Tt)
• 17. Exponential Smoothing with Trend Adjustment
The equation for the trend correction uses a new smoothing constant 
Tt is computed by
where
Tt+1 = smoothed trend for period t + 1
Tt = smoothed trend for preceding period
 = trend smooth constant that we select
Ft+1 = simple exponential smoothed forecast for period t + 1
Ft = forecast for pervious period
• 18. Seasonal Variations with Trend
When both trend and seasonal components are present, the forecasting task is more complex
Seasonal indices should be computed using a centered moving average (CMA) approach
There are four steps in computing CMAs
Compute the CMA for each observation (where possible)
Compute the seasonal ratio = Observation/CMA for that observation
Average seasonal ratios to get seasonal indices
If seasonal indices do not add to the number of seasons, multiply each index by (Number of seasons)/(Sum of indices)