Final Exam
What is the difference between the Auto-Correlation Function (ACF) and the Partial Auto-Correlation Function (PACF)? How would you determine the p and q lags in ARIMA models? [8.5]
ACF plot shows the autocorrelations which measure linear relationship between lagged values of a time series between yt and yt-k PACF measure relationship between yt and yt−k after removing the effects of other time lags PACF → partial correction of a time series with its own lagged values, controlling for the values of the time series at all shorter lags. First partial autocorrelation is identical to the partial autocorrelation because there is nothing between them to remove p= order of the autoregressive part q= order of the moving average part.
Write down the equation of an AR(p), an MA(q), and an ARIMA(p,1,q) model. [8.3-8.5]
AR→ yt= c +(changing parameters-p)yt-1 +et p= order of the autoregressive part c→ constant et → white noise MA → yt=constant+et + (changing parameter-q)et-1 q= order of the moving aveage part Et before changing parameters ARIMA y't= c+p1y't-1 +...+q1e't-1+...+et
How can you capture non-linear relationships in the multiple regression framework? [5.6]
Break functions into piecewise y=f(x)+e, f = possible nonlinear function of x Introduce points where the slope of f can change(bend). These points are called knots
Can forecasts be made in a changing environment? [1.1]
Every environment is changing , and a good forecasting model captures the way in which things are changing. What is normally assumed is that the way in which the environment is changing will continue into the future
How does the information set affect the forecast distribution? [1.7]
Forecast distribution is the set of values that a random variable, and their relative probabilities, could take given what we know from observations/information Equation: yi | I Random variable yi given what we know in I (information)
When would you use a logarithmic transformation? What are the features of a logarithmic transformation? What are some caveats? [2.4, 5.1]
If data show variation that increases or decreases with the level of the series, and also when the data are highly skewed and the outliers make it hard to see what's going on in the bulk of the data Changes in a log value are relative (%) changes on the original scaled Example: log base 10 → increase of 1 log scale corresponds to a multiplication of 10. ii. Log transformations constrain the forecasts to be positive on the original scale. Wt= Log(yt) One caveat is that we cannot take log transformations of predictors with a value of zero (but this can be solved by doing a log-log transformation).
What types of data features can you observe in a time plot? [2.1]
Observation are plotted against time (continuity) Sharp, dips, break and trends
What are confounded predictors? [5.7]
Outside/extraneous variables that correlate with both the dependent and independent variable that explains the correlation between the ID and IV Two variables are confounded when their effects on the forecast variable cannot be separated.
Write down the weighted average form, the component form, and the error correction form of the simple exponential smoothing model! [7.1]
Weighted average form Y hat t+1|t = alpha yt + (1-alpha) y hat t|t-1 Component form Forecast: y hat t+1|t = Lt Smoothing: Lt =alpha yt+(1-alpha)Lt-1 Error correlation Lt= Lt-1+alpha (yt-Lt-1) Lt =Lt-1+alphaLt
What is white noise? How do you recognize that you face white noise? [2.2]
White noise are time series that show no autocorrelation (no repeating patterns) Each autocorrelation is close to zero 95% of spikes in the ACF must lie within 2/square root of T = close to zero If less than 95%, then probably not white noise
Can you use predictor variables in time series forecasting? Explain! [1.4]
Yes. Predictor variables are used in regression to predict another variable. Helps explain what causes variation Error accounts random variation and effects of relevant variables that are not included in the model
How can you detect non-stationarity? How would you deal with non-stationarity? [8.1]
use ACF plot, plot decreases slowly Non stationary data decreases to zero quickly Stationary data will drop to zero quickly. Transformations can stabilize variance of a time-series Random walk is used for non stationary data Longer periods of apparent trends up or down Sudden and unpredictable changes in direction
What are some of the factors affecting the predictability of an event?
-How well we understand the factors that contribute to it -How much data are available -Whether the forecasts can affect the thing we are trying to forecast
Describe the logic behind using scaled errors! How would you interpret a mean scaled error of less than one? How about if it is greater than one? [2.5]
Alternative to using percentage errors when comparing forecast accuracy across series on different scales. Proposed scaling the errors based on the training mean absolute error (MAE) from a simple forecast method. Both the numerator and denominator involve values on the scale of the original data, so the error is independent of the scale of the data. A scaled error is less than one if it arises from a better forecast than the average naive forecast computed on the training data Greater than one → if the forecast is worse than the average naive forecast computed on the training data.
Describe how the seasonal naive method works! When would you use this method? [2.3]
Set each forecast to be equal to the last observed value from the same season of the year ( month, day, etc.) Example,: Feb 2016 value equal Feb 2017 forecasted value One will use this if data is highly seasonal
Describe how cross-validation works! [2.5]
Select observation i for the test set, and use the remaining observations in the training set. Compute the error on the test observation. Repeat the above step for i=1,2,...,N where N is the total number of observations. Compute the forecast accuracy measures based on the errors obtained. Much more efficient use of the available data, as you only omit one observation at each step.
What values can the smoothing parameter take on? What is the impact of various values of the smoothing parameter on the smoothed time series? [7.1]
0 to 1 Subjective manners, previous experience. Estimate them from observed data. Smaller the weight, more weight is given to observations from the past if zero. If one more recent
What is the ten period ahead forecast of the series {0, 1, 2, 3, . . . , 99, 100} using the drift method? When would you use this method? [2.3]
101, 102, 103, 104,105,106,107,108,109,110 Drift method: variation on the naive method that allows the forecasts to increase or decrease over time, where the amount of change over time (drift) is set to be the average change seen in the historical data. Use drift method if you want to forecast based on the historical trend of average change of the data. Sloping up trend
Derive the formula of a centered moving average of order 4 that you could use to smooth quarterly data. Show how you obtain the final weights! [6.2]
2x4-ma T hat t= ½ [1/4 (yt-2+yt-1+yt+yt+1)+ ¼ (y1-1+yt+yt+1+yt+2)] =⅛ yt-2+¼ yt-1+¼ yt+1+⅛ yt+2
What is the one period ahead forecast of the series {7, 5, 6, 4, 2, 3} using the naive method? When would you use this method? [2.3]
3 Used only in time series data All forecasts are simply set to be the value of the last observation Well for many economic and financial time series like stocks Uncertain of likelihood
What is the 90th percentile of the integers {0, 1, 2, 3, . . . , 99, 100}? What is the median? What is the interquartile range? [2.2]
90 (90th percentile is the value for which 90% of the data are smaller) 50 IQR = 75 - 25 = 50 (contains middle 50% of data)
How is a linear trend defined? How is a piecewise linear trend with a bend defined? How are the associated coefficients interpreted? [5.2]
A linear trend can be accounted for by including the predictor x1,t = t A piecewise linear trend with a bend at time t can be specified by including the predictors: x1,t = t and x2,t = { 0 t < τ {(t - τ) t ≥ τ x1,t = β1 = the slope of the trend before time τ; x2,t = β2 → the slope of the trend after time τ is given by β1 + β2
How does multiple regression differ from the simple one? Describe the components of multiple regression? [5.0, 5.1]
A multiple regression has one variable to be forecast (y) and several predictor variables (several x's). A simple regression has one y and one x.
What do we mean by spurious regression? [4.8]
A spurious regression is a type of regression that provides statistical evidence of a linear relationship between independent nonstationary (data does not fluctuate around a constant mean or with a constant variance) variables. Non-stationary Example from the text: data of air passengers in Australia being regressed against rice production in Guinea. High R^2s and high residual autocorrelation can be signs of spurious regression
How do one period ahead forecasts by exponential smoothing differ from a naive forecast or a moving average forecast? [7.1]
Attach larger weights to more recent observation than to observation from the distant past
What are some of the problems with classical decomposition? What other types of decomposition are you familiar with? What are their benefits / disadvantages?
Classical decomposition Additive decomposition Multiplicative Problems The estimate of the trend is unavailable for the first few and last few observations Assumes seasonal component repeats from year to year Longer series this is not reasonable Unusual values from outside factors cannot be detected using classical method. X-12-ARIMA Decomposition: based on classical decomposition but with extra steps and features. Forward in time, backward in time Moving averages Benefit There is no loss of observation at the start and end of the series Seasonal component can vary over time Disadv No r package for X-12-arima Only allows quarterly/monthly data STL decomposition Handle any type of seasonality, seasonal component change over time CANNOT HANDLE TRADING DAY OR CALENDAR VARIATION AND multiplicative decomposition
What are dummy variables? What is the dummy variable trap? How do you interpret the coefficients of dummy variables? Give some examples how you could use dummy variables! [5.2]
Dummy variables (aka indicator variables) are categorical variables that take only two values: "yes" or "no" The dummy variable trap is when someone adds an extra dummy variable to match the total number of categories (this will cause the regression to fail). The last category is specified for the case where the dummy variables are all set to zero. The coefficient associated with a dummy variable is a measure of the effect of that category relative to the omitted category. One example is forecasting daily electricity demand and using the day of the week as a predictor. Another example is evaluating credit score and isolating predictors, such as whether a customer is in full-time employment.
When is time series forecasting appropriate? [1.4]
Forecast something changing over time (estimated using historical observations) Monthly rainfall
What does the coefficient of determination measure? Use a figure to illustrate the concept! Is it a good indicator of the quality of a forecasting model? [4.4, 5.1]
How well a linear regression model fits the data Calculated as the square of the correlation between the the observed y values and the predicted y^(hat) values. r^2 = SSR / SST (yihat-ymean)^2/(yi-ymean)^2 R^2 =1 , predictions are close to the actual R^2=0, prediction are unrelated to the actual values Not a good indicator, it is possible to under or overestimate using this measure. There are no rules of what a good R^2 value. R^2 captures amount of the data Example: Figure 4.3 R= 0.82 thus, 82% of the variation in the carbon footprint is captured by the model
Explain how you would obtain the optimal values of the parameters in exponential smoothing models! How would you initialize the methods? [7.1, 7.6, 7.7]
Initializing the optimal value Specify optimal as the initialized value
What are intervention variables? Give some examples and explain how they are defined! [5.2]
Intervention variables represent instances that may have affected the variable to be forecast. For example, competitor activity, advertising expenditure, industrial action Outlier (spike): when the effect lasts only for one period (use a dummy variable) Dummy variable takes value one for that observation and zero everywhere else Rather than omitting an outlier, you can use a dummy variable to remove its effect. Step: when the intervention causes a level shift in the data (the value of the series changes suddenly and permanently from the time of intervention). This variable takes the value 0 before the intervention and the value 1 from the time of the intervention onward. Intervention variable that causes a permanent change of slope. This intervention is handled using a piecewise linear trend.
How would you determine if your residuals are autocorrelated? [2.2, 2.6]
Look at time plot for residuals Gives idea of what type of pattern lacks in residual Plot the correlogram (aka Autocorrelation function) Run statistical tests (don't need to know equations but understand how they work!) Autocorrelation function (for individual lags) Box-Pierce test and Lujung-Box Large value of Q suggest that autocorrelations do not come from a white noise series. Lack of correlation suggests the forecasts are good
Consider the sample {7, 5, 6, 4, 2, 3}. What is the sample mean? What is the median? What is the standard deviation? What does each of these statistics capture? Explain! [2.2]
Mean:( 2+3+4+5+6+7)/6 = 4.5 The average of the data Median: 2,3,4,5,6,7 → (4+5)/2 =4.5 The middle observation Since the median = mean → equal distribution Standard deviation: [(2-4.5)^2+(3-4.5)^2+(4-4.5)^2+(5-4.5)^2+(6-4.5)^2+(7-4.5)^2]/5 Square root (3.5) → 1.87 The spread/dispersion of data from the mean
What are some caveats you need to keep in mind when using percentage errors? [2.5]
Percentage errors have the disadvantage of being infinite or undefined. assume a meaningful zero. Place heavier penalty on negative errors than on positive errors.
What steps would you follow when fitting an ARIMA model to time series data? [8.7
Plot the data. Identify any unusual observations If necessary, transform the data (Box-cox transformation) to stabilize the variance If the data are non-stationary: take first differences of the data until the data are stationary. Examine the ACF/PACF: is an ar(p) or ma(q) model appropriate? Try your chosen model(s), and use the AICc to search for a better model. Check the residuals from your chosen model by plotting the ACF of the residuals, and doing a portmanteau test of the residuals. If they do not look like white noise, try a modified model. Once the residuals look like white noise, calculate forecast
What are the basic steps in a forecasting task? [1.6]
Problem definition Gathering information (statistical data, expertise....) Preliminary (exploratory) analysis (GRAPHS) Choosing and fitting models Using and evaluating a forecasting model
What is the difference between qualitative and quantitative forecasting? [1.4]
Qualitative forecasting There are no available numerical data, or if the data available are not relevant to the forecasts. Quantitative forecasting Numerical information about the past is available It is reasonable to assume that some aspects of the past patterns will continue into the future Time series data Cross-sectional data
Forecasts are frequently produced for different horizons. Give examples of their use! [1.2]
Short-term forecasts -are needed for the scheduling of personnel, production and transportation. As part of the scheduling process, forecasts of demand are often also required. Medium-term forecasts -are needed to determine future resource requirements, in order to purchase raw materials, hire personnel, or buy machinery and equipment. Long-term forecasts -are used in strategic planning. Such decisions must take account of market opportunities, environmental factors and internal resources.
Why would you use a log-log transformation of your data? How would you interpret the coefficients in a log-log transformed model? [4.7]
Sometimes, there will be instances where data with a non-linear functional form is more suitable than a linear regression. One simple transformation is the log-log transformation, or the (natural) logarithmic transformation. So estimating a log-log functional form produces a constant elasticity estimate in contrast to the linear model which produces a constant slope estimate. log yi = β0 + β1 log xi + εi β1 is the slope (which can also be interpreted as an elasticity). It is the average percentage change in y resulting from a 1% change in x
Describe the diagnostic tools you would use to determine if your model is appropriate! What can we learn from these tools? Why do we care? [5.4]
The difference between the actual value and the fitted values. Ei=yi-yhat How well the graph fits the model. Accuracy
What would you conclude if you found residual autocorrelation? [4.8]
The estimated model violates the assumption of no autocorrelation in the errors, and the forecasts may be inefficient--there is some information left over which should be utilized in order to obtain better forecasts.
Describe the logic of least squares estimation. Use a figure to illustrate the concept. Where are the fitted values? What are the residuals? [4.2, 5.1]
The least squares principle provides a way of choosing β0 and β1 effectively by minimizing the sum of the squared errors. Purple line is used for forecasting The forecast values of y obtained from the observed x values are called "fitted values" (purple line in above figure) ŷ i=β̂ 0+β̂ 1xi, for i=1,...,N Each ŷ i is the point on the regression line corresponding to observation xi The difference between the observed y values and the corresponding fitted values are the "residuals" ei=yi−ŷ i ei=yi−β̂ 0−β̂ 1xi.
How would you construct an odd order moving average of a time series? What is the effect of changing the order of the moving average? [6.2]
The order of the moving average determines the smoothness of the trend-cycle estimate. Odd orders are symmetric, in a moving average of order m =2K+1 Or m-MA A larger order means a smoother curve If the order is even, then there is no symmetric 2xm-ma
There are two types of quantitative forecasting depending on the type of available data. What are these data types? Describe their distinguishing features! [1.4]
Time series data Useful when you are forecasting something that is changing over time Observed sequentially over time and estimate how observation will continue to future. Cross-sectional data Predict the value of something we have not observed by using information from cases that we have observed
Describe what we use hypothesis testing for! What is a P-value? How do you use P-value in hypothesis testing? [4.6]
To examine whether there is enough evidence to show that x and y are related Testing whether the predictor variable x has had an identifiable effect on y. If x and y are unrelated, then the slope parameter β̂ 1 = 0 P-value size The smaller the P-value is (or equal to significant level) , the more unlikely to have arisen if the null hypothesis were true HO : β̂ 1 = 0
What are the three primary time series patterns? How do they differ from each other? [6.1]
Trend long-term increase or decrease in the data. Seasonal A seasonal pattern exists when a series is influenced by seasonal factor Cyclic A cyclic pattern exists when data exhibits rises and falls that are not of fixed period. Fluctuation is usually of at least 2 years
What are the three basic time series patterns? What are their distinguishing features? [2.1]
Trend Long-term increase or decrease in the data. Example: overall long-term increase in antidiabetic drug sales in Australia Seasonal Time series affected by seasonal factors such as year or the day of the week (fixed and known time length) Example: monthly Cycle When the data exhibit rises and falls that are not a fixed period (variable and unknown time length) Example: business cycle
When is cross-sectional forecasting appropriate? [1.4]
Use information from cases that we have observed example:Fuel economy data for a range of 2009 model cars. We are interested in predicting the carbon footprint of a vehicle not in our data set using information such as the size of the engine and the fuel efficiency of the car. Used when the variable to be forecast exhibits a relationship with one or more other predictor variables.
Are the correlation coefficient and the regression coefficient related? Explain! [4.3]
Yes, correlation coefficient ( r ) is needed to find the regression coefficient r(sy/sx) sy and sx are standard deviation of y and x observations. The ratio (sy/sx) is needed and demonstrates that x predicts y
How would you obtain genuine forecasts from historical data? Would you try to achieve a "perfect fit"? Why? [2.5]
he accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common to use a portion of the available data for fitting, and use the rest of the data for testing the model (split data into training set and test set) Perfect fit → using a model with enough parameters 20% of total sample Size of the test set should ideally be at least as large as the maximum forecast horizon required. Over fitting a model to data is as bad as failing to identify the systematic pattern in the data
Why do we look at residual plots? Are outliers and influential observations the same? How would you handle outliers? [4.4]
residual is the unpredictable random component of each observation A simple and quick way for a first check is to examine a scatterplot of the residuals against the predictor variable Observations that take on extreme values compared to the majority of the data are called "outliers" Observations that have a large influence on the estimation results of a regression model are called "influential observations" A non-random pattern may indicate that a nonlinear relationship may be required, or some heteroscedasticity is present (i.e., the residuals show non-constant variance), or there is some left over serial correlation (only when the data are time series)
Describe the components of a simple linear model! Use a figure to illustrate the components. Do we make any assumptions about the errors? [4.1]
y=β0+β1x+ε The parameters β0 and β1 determine the intercept and the slope of the line respectively intercept β0 represents the predicted value of y when x=0 The slope β1represents the predicted increase in Y resulting from a one unit increase in x random "error", εi Assumptions have mean zero; otherwise the forecasts will be systematically biased. are not autocorrelated; otherwise the forecasts will be inefficient as there is more information to be exploited in the data. are unrelated to the predictor variable; otherwise there would be more information that should be included in the systematic part of the model.
Write down the matrix formulation of a multiple regression equation! [5.5
yi= bo+b1x1,i+Bkxk,i+ei
What can you use scatterplots and scatterplot matrices for? [2.1]
Scatterplots: Examining relationships between variables. A scatterplot helps one visualize the relationship between the variables Scatterplot matrices: examine relationship for several potential predictor variables. Useful to plot each variable against each other variable.
Can you describe a simple method to calculate a prediction interval? Are there any limitations of this method? Why do we care about prediction intervals? [2.7]
ŷi ± kσ̂ (k is the multiplier) Assumes that the residuals are uncorrelated and normally distributed (σ̂ is the standard deviation of the residuals of the data) Since a prediction interval gives us an interval within which we expect yi to lie (with a specific probability), using a prediction interval expresses the uncertainty of the forecast
Correlation is not causation. Explain! [5.7]
Variable x may be useful for predicting a variable y, but that does not mean x is causing y. It is possible that x is causing y, but it may be that the relationship is more complicated. Two variables can be correlated but not cause each other
What is the one period ahead forecast of the series {7, 5, 6, 4, 2, 3} using the average method? When would you use this method? [2.3]
4.5 What is it? The forecasts of all future values are equal to the mean of the historical data. Can be used in predicting a value not in the data (cross-sectional data). The prediction for values not observed is then the average of those values that have been observed.
What are some desirable properties of residuals? Why are the properties you listed desirable? [2.6]
A good forecasting method will yield residuals with the following properties The residuals are uncorrelated. If there are correlations between residuals, then there is information left in the residuals which should be used in computing forecasts The residuals have zero mean. If the residuals have a mean other than zero, then the forecasts are biased Two properties that are useful but not necessary are that the residuals have constant variance and are normally distributed. These properties make the calculation of prediction intervals easier.
Define stationary time series! Give examples of types of non-stationarity! [8.1]
A stationary time series is one whose properties do not depend on the time at which the series is observed Time series with trends, or with seasonality are not stationary White noise series is stationary Look the same at any given point Roughly horizontal
What do we mean by time series decomposition? What do we mean by seasonal adjustment? [6.1]
A statistical method that deconstructs a time series into several components, each representing one of the underlying categories of patterns: Seasonal component, a trend-cycle component, remainder component. When the seasonal component is removed from the original data yt-St Multiplicative data seasonally adjusted values are obtained using yt/ST.
Describe the steps of the classical additive decomposition! How do they differ from the classical multiplicative decomposition? [6.3]
Additive decomposition Step 1: if m is an even number, compute the trend-cycle component using a 2xm-ma to obtain T hat t. If m is an odd number, compute the trend cycle component using an m-Ma to obtain T hat t. Step2: Calculate the detrended series: yt-T hat t. Step 3: estimate the seasonal component for each month by simply averaging the detrended values for that month. S hat t Step 4: the remainder component is calculated by subtracting the estimated seasonal and trend-cycle components E hat t= yt-(T hat t) - (S hat t) In multiplicative decomposition Subtractions are replaced by divisions Step 2: calculate detrended series: yt/T hat t Then, step 4 Ehat t=yt/T hat tS hat t
What methods would you use to produce ex-ante forecasts from regression models? How do these differ from ex-post forecasts? What can you learn from comparing the two approaches? [4.8, 5.2]
Ex-ante forecasts use only information that is already available at the time of the forecast. For example, ex-ante forecasts of consumption for the four quarters in 2011 should only use information that was available before 2011. Ex-post forecasts use later information on the predictors (information after the forecast). For example, ex-post forecasts of consumption for the four quarters in 2011 would use the actual information from each of those four quarters in 2011. Ex-ante forecasts are genuine forecasts since we are making actual predictions based on all available information at the time we are making the forecast. Ex-post forecasts are useful for studying the behavior (and possibly the accuracy) of forecasting models. Evaluating both types can help to separate out the sources of forecast uncertainty. This will show whether forecast errors have arisen due to poor forecasts of the predictor or due to a poor forecasting model.
Why is forecasting a subdiscipline of statistics? [1.7]
Forecasting uses probability to estimate/predict future outcomes that are unknown (random variable). Example: prediction interval (range) → 95% interval is commonly used in statistics
What is the difference between forecasting, planning, and goals? [1.2]
Forecasting: predicting the future as accurately as possible using historical data and of future events Goals: What you would like to happen.Too often, goals are set without any plan for how to achieve them, and no forecasts for whether they are realistic. Planning: A response to forecasts and goals. Determining the appropriate actions that are required to make your forecasts match your goal
Describe the gist of exponential smoothing. [7]
Forecasts produced using exponential smoothing methods are weighted averages of past observations, with the weights decaying exponentially as the observations get older. In other words, the more recent the observation the higher the associated weight. This framework generates reliable forecasts quickly and for a wide spectrum of time series which is a great advantage and of major importance to applications in industry.
Describe some methods you could use to select predictors in your model! Are there any conventionally used methods that you would not use? Why are the methods you are suggesting more appropriate than the methods you advise against? [5.3]
It is recommended to plot the forecast variable against a particular predictor and to do a multiple linear regression on all the predictors used in a model. A conventionally used method that is NOT recommended is dropping a particular predictor if it shows no noticeable relationship with the forecast variable. Another conventionally used method that is NOT recommended is doing a multiple linear regression on all the predictors and then disregard all variables whose p-values are greater than 0.05. Dropping a particular predictor if it shows no noticeable relationship is invalid because it's not always possible to see the relationship straight from the scatterplot, especially when the effect of other predictors has not been accounted for. Disregarding all variables whose p-values are greater than 0.05 is also invalid because p-values can be misleading when two or more predictors are correlated with each other and because statistical significance does not always indicate predictive value. Use a measure of predictive accuracy: Adjusted R2, cross-validation, Akaike's Information Criterion (AIC), corrected Akaike's information criterion (AICc) → maximize adjusted R2 or minimize CV, AIC, AICc Schwarz Bayesian information criterion is not the best measure of predictive accuracy because it assumes there is a true underlying model that will be selected with enough data.
How would you calculate the mean absolute error and root mean squared error of your forecasts? Can you compare these across data sets? [2.5]
Mean Absolute error: Forecast error ei = yi -yhati (same scale as the data) MAE = mean (|ei|) Root mean squared error RMSE = Square root [mean (e^2i)] No you cannot compare across data set because accuracy measures that are based on ei are scale-dependent.
What does the correlation coefficient measure? Interpret some possible values of the correlation coefficient. Discuss any caveats related to the correlation coefficient! [2.2]
Measure the strength of the relationship between two variables in a linear relationship Ranges from -1.00 to 1.00. Linear relationship must be present
What does autocorrelation mean? What do different orders of autocorrelation indicate? What is an autocorrelation function? [2.2]
Measures the linear relationship between lagged values of time series. It is the similarity between observations as function of the time lag between them. Tool for finding repeating patterns. r1= measures the relationship between yt and yt-1 r2= yt and yt-2 and so on yt-k, for different values of K Compares the relationship between lagged values of time series for each variable.
When and how would you incorporate trading days and distributed lags into your model? [5.2]
Trading days can be incorporated into the model when dealing with sales data, since the number of trading days can vary from month to month X1=# Mondays in month X2=# Tuesdays in month etc. advertising expenditure can have distributed lags as the effect of advertisement can last beyond the actual campaign. x1= advertising for previous month
What types of plots can you use to analyze seasonal patterns? What do they display? [2.1]
Use seasonal plots when a seasonal pattern is visible and important to forecasting.Seasonal patterns allow ones to analyze patterns, overlap, and changes. Time of the year of the day of the week is stated
How would you calculate the standard error of the regression? What does it measure? [4.4]
Use standard deviation equation to calculate, however, instead of N-1 use N-2. This is because two parameters are involved in computing residual Intercept Slope compare error to the sample mean of y or its standard deviation to gain perspective on the accuracy of the model Measures how well the model has fitted the data.
How would you obtain a forecast from a simple linear model? Would you expect the forecast interval to have uniform width for all values of the explanatory variable? [4.5]
Use the regression line equation ŷ i=β̂ 0+β̂ 1xi. Input value x into the equation and obtain a forecasted yhat No you would not expect uniform width for all values of the explanatory variables. X that is farther from X mean = wider
Why and how would you use a population adjustment? [2.4]
Use when population changes over time because population may affect data Used for data affected by population. Better option than total. Per-capita data Example: number of hospital beds in a particular region over time. Real increase in the number of beds? Or increase in population?
Why and how would you use an inflation adjustment? [2.4]
Used when data is affected by the value of money. Are best adjusted before modelling Example: Average cost of a new $200,000 house have increased over the years Price index is used. (Zt) xt=yt/Zt *zyear yt= the original value.
What are some of the issues when determining what to forecast? [1.3]
What are forecast needed for in manufacturing environment: every product line, or for groups of products? every sales outlet, or for outlets grouped by region, or only for total sales? weekly data, monthly data or annual data? Will forecasts be required for one month in advance, for 6 months, or for ten years? How frequently are forecasts required? Talking to people who will use the forecasts to understand their needs, how forecasts are used,
What is multicollinearity? How does it arise and what are its consequences? [5.7]
When similar information is provided by two or more of the predictor variables in a multiple regression Two predictors are highly correlated with each other. Knowing the value of one variable tells you a lot about the value of of the other variable. Similar information Linear combination of predictors is highly correlated with another linear combination of predictors. Knowing first group makes you know the second group Consequences If there is perfect correlation, it is not possible to estimate the regression model If there is a high correlation then the estimation of the regression coefficient is computationally difficult. The uncertainty associated with individual regression coefficients will be large. Difficult to estimate Forecasts will be unreliable if the values of the future predictors are outside the range of the historical values of the predictor
Why and how would you use a calendar adjustment? [2.4]
When there will be variation in the data between months or seasonal variation Example: studying monthly milk production on a farm Looking at average daily production instead of average monthly production, we effectively remove the variation due to the different month lengths Simpler patterns are usually easier to model and lead to more accurate forecasts. Adjustment can be done for sales data when the number of trading days in each month will vary.
