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IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
IBM401 Lecture 12
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IBM401 Lecture 12


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  • 1. Quantitative Analysis for Business
    Lecture 12 – Final Exam Review
    October 4th, 2010
  • 2. Final exam
    No. of questions:
    3 hours
    50% of final grade
  • 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.
  • 10. Adjusted R2
    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
    n = number of observations
    k = number of independent variables
    Ra2 = adjusted R2
  • 11. Adjusted R2
    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
    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
    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
    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
    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)