Chapter 4 and 5 quantitative methods of business

Regression analysis

- very valuable tool for a manager Understand the relationship between variables Predict the value of one variable based on another variable

Forecast error

= Actual value - Forecast value

more responsive

A high value of B makes the forecast ---to changes in trend

Forecasting - Trend and Random

A more complex model can be used The basic approach Develop an exponential smoothing forecast Adjust it for the trend

(errors)

A plot of the residuals ----- often highlights glaring violations of assumptions

Cautions and Pitfalls

A t-test for the intercept (b0) may be ignored as this point is often outside the range of the model A linear relationship may not be the best relationship, even if the F test returns an acceptable value A nonlinear relationship can exist even if a linear relationship does not Even though a relationship is statistically significant it may not have any practical value

Exponential smoothing

A type of moving average Easy to use Requires little record keeping of data

low value

A----- of B gives less weight to the recent trend and tends to smooth out the trend

Sales Force Composite

Allows individual salespersons estimates Reviewed for reasonableness Data is compiled at a district or national level

Correlation Coefficient

An expression of the strength of the linear relationship Always between +1 and -1

average season higher than average lower than average

An index of 1 indicates an An index > 1 indicates the season is An index < 1 indicates a season

model building

As the number of variables increases, the adjusted r2 gets smaller unless the increase due to the new variable is large enough to offset the change in k

Jury of Executive Opinion

Collects opinions of a small group of high-level managers May use statistical models for analysis

Measures of Forecast Accuracy

Compare forecasted values with actual values See how well one model works To compare models

Common qualitative techniques

Delphi method Jury of executive opinion Sales force composite Consumer market surveys

Assumptions of the Regression Model

Errors are independent Errors are normally distributed Errors have a mean of zero Errors have a constant variance

variance

Estimated using the mean squared error (MSE), s2

dependent

Explanatory or predictor variable

Multiple Regression Analysis

Extensions to the simple linear model

Time-Series Models

Extrapolations of past values of a series

Trend Projections

Fits a trend line to a series of historical data points

Components of a Time Series

Four possible components Trend (T) Seasonal (S) Cyclical (C) Random (R)

low

If the F statistic is large, the significance level (p-value) will be , - unlikely would have occurred by chance

Cautions and Pitfalls

If the assumptions are not met, the statistical test may not be valid Correlation does not necessarily mean causation Multicollinearity makes interpreting coefficients problematic, but the model may still be good Using a regression model beyond the range of X is questionable, as the relationship may not hold outside the sample data

rejected

If the null hypothesis can be ----, we have proven there is a relationship

useful

If there is very little error, MSE would be small and the F statistic would be large - model is

Time-Series Models

Ignores factors such as Economy Competition Selling price

collinear

In some cases variables contain duplicate information When two independent variables are correlated, they are said to be

nonlinear regression

In some situations, variables are not linear Transformations may be used to turn a nonlinear model into a linear model

Qualitative Models

Incorporate judgmental or subjective factors Useful when subjective factors are important or accurate quantitative data is difficult to obtain

Consumer Market Survey

Information on purchasing plans solicited from customers or potential customers Used in forecasting, product design, new product planning

Delphi Method

Iterative group process Respondents provide input to decision makers Repeated until consensus is reached

Trend Projections

Linear model developed using regression analysis is simplest

Measure of accuracy

Mean absolute deviation (MAD):

Measures of Forecast Accuracy

Mean squared error (MSE) Mean absolute percent error (MAPE) Bias is the average error

Multiple Regression Analysis

Models with more than one independent variable

Forecasting Random Variations

No other components are present Averaging techniques smooth out forecasts Moving averages Weighted moving averages Exponential smoothing

Testing the Model for Significance

Performing a statistical hypothesis test

Time-Series Models

Predict the future based on the past

Trend Projections

Projected into the future for medium- to long-range forecasts

simple linear regresssion

Random error

Seasonal Variations

Recurring variations over time may indicate the need for seasonal adjustments in the trend line

Main purpose of forecasting

Reduce uncertainty and make better estimates of what will happen in the future

simple linear regresssion

Regression models used to test relationships between variables

test statistic

Reject the null hypothesis if the ----- is greater than the F value from the table in Appendix D. Otherwise, do not reject the null hypothesis:

level of significance

Reject the null hypothesis if the observed significance level, or p-value, is less than the ----(). Otherwise, do not reject the null hypothesis:

Three measures of variability

SST - Total variability about the mean SSE - Variability about the regression line SSR - Total variability that is explained by the model

Subjective methods

Seat-of-the pants methods, intuition, experience

Selecting the Smoothing Constant

Selecting the appropriate value for alpha is key to obtaining a good forecast The objective is always to generate an accurate forecast The general approach is to develop trial forecasts with different values of alpha and select the alpha that results in the lowest MAD

Components of a Time Series

Sequence of values recorded at successive intervals of time

Multiple Regression Models

Similar to simple linear regression models The p-value for the F test and r2 interpreted the same The hypothesis is different because there is more than one independent variable The F test is investigating whether all the coefficients are equal to 0 at the same time

significance

Testing the model for ----- helps determine if the values are meaningful

b1 dont equal 0

The alternate hypothesis is that there is a linear relationship

model building

The best model is a statistically significant model with a high r2 and few variables As more variables are added to the model, the r2 value increases For this reason, the adjusted r2 value is often used to determine the usefulness of an additional variable The adjusted r2 takes into account the number of independent variables in the model

r2.

The coefficient of determination is

new variable

The easiest approach - develop a

Exponential Smoothing with Trend

The equation for the trend correction uses a new smoothing constant B Ft and Tt must be given or estimated Three steps in developing FITt 1, Compute smoothed forecast Ft+1 2Update the trend 3Calculate the trend-adjusted exponential smoothing forecast (FITt +1)

quadratic model

The nonlinear model is a

dummy variables

The number of ---- must equal one less than the number of categories of the qualitative variable

The coefficient of determination

The proportion of the variability in Y explained by the regression equation

Multiple Regression Models

The test statistic is calculated and if the p-value is lower than the level of significance (), the null hypothesis is rejected

Trend Projections

Trend equations can be developed based on exponential or quadratic models

simple linear regression

True values for the slope and intercept are not known Estimated using sample data

Time-Series Models

Two basic forms Multiplicative Demand = T x S x C x R Additive Demand = T + S + C + R Combinations are possible

Moving Averages

Used when demand is relatively steady over time The next forecast is the average of the most recent n data values from the time series Smooths out short-term irregularities in the data series

Time-Series Models

Uses only historical data on one variable

dependent

Value depends on the value of the independent variable(s)

trial-and-error approach

Values are often selected using a ---based on the value of the MAD for different values of

dependent variable or response variable

Variable to be predicted is called the

Testing the Model for Significance

We use the F statistic

multicollinearity

When --- is present, hypothesis tests for the individual coefficients are not valid but the model may still be useful

multicollinearity

When more than two independent variables are correlated,-----exists

reject the null hypothesis

When the F value is large, we can ------and accept that there is a linear relationship between X and Y and the values of the MSE and r2 are meaningful

MSE and r2

When the sample size is too small, you can get good values for ---- even if there is no relationship between the variables

ANOVA table

With software models, an --- is typically created that shows the observed significance level (p-value) for the calculated F value This can be compared to the level of significance () to make a decision

Errors

are assumed to have a constant variance (Q 2), usually unknown

Binary (or dummy or indicator) variables

are special variables created for qualitative data

Backward stepwise regression

begins with all the independent variables and deletes the least helpful

Regression models

can be developed for any variables X and Y

Describes an F distribution with

degrees of freedom for the numerator = df1 = k degrees of freedom for the denominator = df2 = n - k - 1

Exponential smoothing

does not respond to trends

Multiple regression models

have more than one independent variable

A seasonal index

indicates how a particular season compares with an average season

bo

intercept (value of Y when X = 0)

alpha

is a weight (or smoothing constant) with a value 0 ≤ alpah ≤ 1

A dummy variable

is assigned a value of 1 if a particular condition is met and a value of 0 otherwise

Regression analysis

minimizes the sum of squared errors

Simple linear regression

models have only two variables

Independent variable

normally plotted on X axis

Dependent variable

normally plotted on Y axis

k =

number of independent variables

n =

number of observations in the sample

Scatter diagram or scatter plot

often used to investigate the relationship between variables

With average error

positive and negative errors cancel each other out

A forward stepwise procedure

puts the most significant variable in first, adds the next variable that will improve the model the most

e

random error

b1

slope of the regression line

Stepwise regression

systematically adds or deletes independent variables

b1=0

the null hypothesis is that there is no relationship between X and Y

Weighted moving averages

use weights to put more emphasis on previous periods Often used when a trend or other pattern is emerging