FRL 4631 MT 1
Forecast with Holt's Model: example
Forecast: Ft+1 = aXt + (1-a)(Ft + T t) Trend: •Tt+1 = g(Ft+1 -Ft) + (1-g)Tt •H t+m = Ft+1 + mTt+1
Simple smoothing formula
Ft+1 = a( Xt ) +(1- a ) F t •Ft+1 = Forecast value for period t+1 •a= smoothing constant (weight) (0< a <1) •Xt=actual value at period t •Ft= Forecast value for period t
when there is trend in the data, which of the following methods provide the best forecast.
Holt Model
a correlation coefficient of 0.86 between Y the dependent variable and X1 an independent variables implies:
X1 is a good candidate to be added to the regression, but we should also consider the causality between Y and X1.
Third quick check what is the value of R2
•If R2 is close to 1, the set of the independent variables can explain the variation in the dependent variable. •If R2 is close to 0, the set of the independent variables can not explain the variation in the dependent variable.
Holt's model
is used when there is an upward or downward trend in the data
Winter's model
is used when there is both trend and seasonality in the data.
If the series is smooth, value of alpha
is usually close to 1.
Simple exponential smoothing
uses the past values of a time series to forecast the future value of the same time series.
If the series has a great deal of random variation
value of a is usually close to 0
a p value (probability) of inventory implies that:
we can reject the following null hypothesis, so we should keep inventory in the regression H0 : β inventory = 0,
the T value of newcap suggests:
we can reject the following null hypothesis, so we should keep newcap in the regression H0 : βnewcap = 0
to test the null hypothesis that H0 : b1=b2=b3=b4=b5=0
we can use F test with 5 , 449 degree of freedom.
Problems with R2
• Adding new independent variables to the regression always will increase R2, even if the variable is irrelevant.
Population and sample Regression lines
• If we have all the possible data, we can estimate the population regression line , which gives us the correct values of intercept and slope. However, In practice we don't have that information. We usually have to work with a sample of data.
Basic Statistical Evaluation of Regression Method
•Does the model makes economic sense? ( are the coefficients signs correct). •Is there a statistically significant relationship between dependent and independent variables? (t test) •What percentage of variation in the dependent variable is explained by variation in the independent variable?
Comparing regressions
•Don't compare the R2 of two regressions if •1) the dependent variables are not the same in the two regression. •2) the number of independent variables are not the same. •3) the functional forms are not the same.
F statistics
•F test is a test of overall significance of the estimated multiple regression. •H0: b1 = b2 = ...= bk =0 • or • H0: R2 = 0 • •The null hypothesis is rejected if the calculated F from the data is greater than the critical value of the F-distribution at 5% or 10%. F test is always one sided.
First quick checkDoes the model make economic sense
•Find out what the economic theory or business logic say about the relationship between the dependent and the independent variable. For example, risk is known to have a negative effect on returns.
Holt-Winter model equations seasonal data
•Ft = a(Xt / St-p ) + (1-a)* ( Ft-1 +T t-1) smoothed value •Tt = g*( Ft - Ft-1) + (1- g) Tt-1 Trend value •St = b*( Xt/Ft ) + (1- b)* St-p seasonal index Where , Ft is the smoothed value for time t, Tt is the trend estimate for period t, St is seasonality estimate. alpha gamma, beta are the smoothing constants and 0< alpha, gamma, beta <1 P= is the number of periods in the seasonal cycle, sop=4 for quarterly data and p=12 for monthly data.
the correlation coefficient between shipment and inventory is
0.61
Use the car sales data and apply the Holt model. The one period forecast of the car sale is
1109.99
Use industrial production data and apply the simple smoothing method. What is the one period ahead forecast.
111.87
Use industrial production data and apply the simple smoothing method. What is the average error of the forecast.
2.25
The critical value to test the following null hypothesis at 5% is: H0: R2 = 0
2.28
Use the data of housing sales and apply the model for trend. What is the forecast of the sales of the second quarter of 2021 (two period ahead)
315078.2
Use the data of housing sales and apply the model for trend and seasonality . What is the forecast of the sales of the first and the third quarter of 2021.
334019.8, 296114.2
you would like to forecast housing sales for future. using the data of housing sale choose one of the following statements:
All of the above is correct.
Regression model disturbances (forecast errors)
All of the options are correct.
The reported p values (probabilities) indicate that :
All the independent variables are statistically significant at 5% level.
Which of the following would indicate a perfect model fit
R2 = 1
Multiple regression ex
Sales = 16.41 - 8.25 price +0.59 adv • If price increases by one unit, sales will decrease by 8.25 units, Assuming all other variables remain constant. • If Adv increases by one unit, sales will increase by 0.59 units, assuming all other variables remain constant.
The data file Manufactures contains data from the U.S. Census Bureau's American Survey of Manufactures on 455 industries in 1994. The variables are as follows: shipment = value of output shippep. materials = value of materials inputs used in production.Newcap = expenditure on new capital by this industry. Inventory = value of inventory hold. Managers = number of supervisory workers employed. Workers = number of production workers employed. The first four variables are measured in thousands of dollars. Estimate the regression of Shipments = b 0 + b 1 managers + b 2 workers + b 3 materials + b 4 newcap +b 5 inventory + ε i
The coefficient of determination indicates that 98 percent of variation in shipments is explained by the regression.
here is an approximate linear relationship between the height of females and their age (from 5 to 18 years) described by: height = 50.3 + 6.01(age) where height is measured in cm and age in years. Which of the following is not correct?
The estimated intercept is 50.3 cm which implies that children reach this height when they are 50.3/6.01=8.4 years old.
use the car sale data which shows the monthly car sale of a company from Jan 2016 to december of 2020 and choose of the following answeres
The pattern of data show trend.
you would like to forecast the industrial production of a country. Use the data set "indprod" which contains the past data and determine what type of smoothing method you would use to forecast the industrial production of January 2021.
The pattern of data shows random behavior with no trend or seasonality
If the correlation coefficient between Y and X in an ordinary least squares regression = 1.00, then
all the data points must fall exactly on a straight line with a positive slope.
drop newcap and inventory and Estimate the following regression: Shipments = b 0 + b 1 managers + b 2 workers + b 3 materials + ε i
dropping these two variables make the results worse because Akaike criteria increases.
difference between r^2 and adjusted r^2
r^2 increases where as adjusted r^2 doesn't increase
Akaike information criterion (AIC) and schwarz criterion
shows if you should add variables or not
Testing the null hypothesis that the slope coefficient is zero uses what sampling distribution for small sample sizes?
t distribution with n-k-1 degrees of freedom.
use the quarterly housing sale data which shows the housing sales for a county in Claifornia for the period 2011 Q!- 2020Q4 and choose of the following answeres
the Pattern of data show seasonality.
Rule Akaike information criterion (AIC) and schwarz criterion decrease
the lower they are the better
Holt's Model
•In some situations, the observed data show an upward or downward trend. •When a trend exists, we can improve the accuracy of our forecast by including the trend information in our estimation of forecast. •To forecast time series with a trend, we select the Holt's model. •Holt's two-parameter exponential smoothing method is an extension of simple exponential smoothing.
Holt-Winter model multiplicative- seasonal data
•Many of monthly and quarterly time series show a seasonal pattern. •In these cases we can improve the accuracy of the forecast by adjusting it for the seasonal movements. •Holt-Winter's model is the second extension of the simple smoothing model. It is used for data that shows both trend and seasonality. •In the Holt-Winter model we add a third equation to the Holt's model to adjust for the seasonality.
Covariance
•Measure the linear relationship between two variables •If the sign of covariance is positive, the two variables move in the same direction. If the sign is negative the two variables move into opposite directions. •The magnitude of covariance changes with the unit of measurement and therefore does not have any use. •If x and y are independent of each other, then the cov(x,y)=0. •If cov(x,y) =0, then x and y are not necessarily independent. All we can say is that a linear relationship does not exist between them
Correlation Coefficient
•Measures the linear relationship between two variables. •The sign of cor(x,y) shows the direction of linear association. •The magnitude of correlation is always between +1 and -1, and show the strength of the relationship. •A low correlation (close to zero) only means that a linear relationship doesn't exist . A nonlinear relationship might exist. •Remember that neither correlation nor covariance is an indicator of causation between X and Y.
Exponential smoothing
•Most managers have to forecast the inventories or sales of many product each day, week, or month. •While sophisticated forecast method can be used, some simple method of time series smoothing will do the job in this case. •In smoothing method we use a form of weighted average of the past data to smooth (eliminate) the short term fluctuations.
Multiple regression
•Mr. Bump wants to improve his weekly sale volume. He decides to investigate the effects of advertising expense and price on his sales volume. •Before running a regression, it is helpful to look at the correlation coefficients between sales and the two independent variables: advertising expense and price.
Sources of error
•Omitted variables •Measurement error •Wrong functional form •Unpredictable or purely random variation in human behavior.
Adjusted R2
•R2 will increase with each additional explanatory variable, because of it's formula: •Usually when we add a new variable to the regression, due to OLS technique, Se2 will decrease and R^(2 )will increase . •As more variables are added to the model, Σe2 decreases, however k =number of x's increases, and N-k decreases too, so adjusted R2 can either increase or decrease depending how much the new variable can reduce the forecast error.
Third quick checkHow much of the variation of Y is explained by X?
•R^2 (coefficient of determination) shows what percent of the variations (up and down movement) in the dependent variable (Y) is explained by variation in the independent variable (X).
What is Regression Analysis
•Regression analysis helps us to estimate the mathematical relationships between the variables. •We all know that if we increase advertising, sales would increase. But usually we want to find out exactly how much sales will increase if we increase advertising expenditure by $1000.
Interpretation
•Slope , b1, shows how many units Y would change if X changes by one unit. •Intercept, b0 , shows the value of Y if X=0, or it simply shows the height of the regression line. •Suppose we estimated the following regression using the OLS method: •Y = 4.2 + 3.5 X •Slope b1= 3.5. It implies if we increase X (our independent variable) by one unit, Y would increase by 3.5 units. •Intercept b0= 4.2, shows Y would be 4.2 when X=0, or simply the height of regression is 4.2.
Applications of simple regressionForecasting with a simple linear Trend.
•Sometimes we can make reasonable forecast by a simple time trend regression. •Y = b0 + b1 t • where Y is the variable that we want to forecast and t is a simple time trend
Important point
•Suppose Y is the actual value. •error e = Y - b0 + b1 X • Since the population information is not usually available, e, population error term is not observable. •sample residual, e, is observable and can be measured as e = Y - b0 + b1 X
Regression Analysis ex
•Suppose we are interested to estimate sales as a function of advertising •sale = b0 + b1 advertising •sales = 425 +7.2 advertising •Now suppose we want to forecasts sales if advertising is $1000 •Forecast of Sales = 425 +7.2*1000 =7625
Forecast with simple smoothing
•The Forecast of X for m period ahead of time t is estimated by •H t+m = Ft+1 where: m = number of period ahead to forecast.
Use of the Regression Analysis
•The description of economic theory •The testing of hypothesis about economic theory •The forecasting of future economic activity.
Advantage of exponential Smoothing Models
•The major advantage of exponential smoothing algorithms is their ability to produce quite reliable forecasts relatively quickly for a large collection of time series. This is particularly valuable as an input to inventory management, for which monthly, or perhaps quarterly sales forecasts are needed.
Important point
•The regression equation must always be linear in coefficient , otherwise we can not estimate it. The power of independent variables X, however, can be more than one. •Y = b0 + b1 X1^3 + b2 X2^4 +e o.K. •Y = b0 + b1^3 X1 + b2^4 X2 + e not O.K.
Akaike and Schwarz statistics
•There are two other model-specification statistics that can be of use in selecting the correct independent variables. •These are the Akaike Information Criterion and the Schwarz Criterion. •The Akaike and Schwarz statistics are minimized in the best model. •These statistics are constructed so that, There is a penalty for increasing the number of independent variables. The variable must be added to the regression only if it can add enough accuracy in estimation to offset the penalty and reduce Akaike and Schwarz statistics.
Regression Analysis
•To find the mathematical relationship between X and Y, we start by assuming that Y is a linear function of X: • Y = b0 + b1 X •Y= dependent variable because its value depends on X, •X=independent variable because its value does depend on Y Y = b0 + b1 X •b0 is the intercept of the line. •b1 is the slope of the line. •b0 and b1 are called the parameters or coefficients of the regression. •Causality should always go from X to Y, and not from Y to X.
How to estimate regressions
•We estimate the regression line such that ∑e2 is minimized. •the estimation method is called the ordinary Least Squared (OLS)
Holt-Winter Forecast equation
•Wt+m = (Ft +mTt) St+m-p
Regression analysis ex
•Y = b0 + b1 X+ e • where Y =actual value, •and b0 + b1 X = forecast. •e = forecast error = actual value -forecast •error term e is a random variable which takes positive and negative values randomly.
Simple smoothing
•is used when there is no trend or seasonality present in the data.