Lecture 8:Review: Forecasting

methodsUnderstanding Markets

and Industry Changes

The full model

The model with seasonality, quadratic trend, and ARMA components can be written:

Dummy variables, Djt control for seasonality.

The variable t controls for trend if your data appears to grow or fall at a linear rate.

If the “trend” doesn’t look like a straight line, you may consider taking the natural log of your original series, or including the quadratic trend part, t2.

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y t = b1D1,t + ...+ bsDs,t +a1t +a2 t 2 + ut ,

ut = φ1ut−1 + φ2ut−2 + ...+ φput− p + ...+

et + θ1et−1 + ...+ θqet−q

The full model continued…

The model with seasonality, quadratic trend, and ARMA components can be written:

If your data is related to past observations of itself, include autoregressive components (expect processes having autoregressive components to have slowly decaying autocorrelations).

If your data is related to past errors made in fitting the data, include moving average components (expect processes having moving average components to have slowly decaying partial autocorrelations).

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y t = b1D1,t + ...+ bsDs,t +a1t +a2 t 2 + ut ,

ut = φ1ut−1 + φ2ut−2 + ...+ φput− p + ...+

et + θ1et−1 + ...+ θqet−q

Model selection

An important statistic that can used in choosing a model is the Schwarz Bayesian Information Criteria. It rewards models that reduce the sum of squared errors, while penalizing models with too many regressors.SIC=log(SSE/T)+(k/T)log(T), where k is the number

of regressors.

The first part is our reward for reducing the sum of squared errors. The second part is our penalty for adding regressors. We prefer smaller numbers to larger number (-17 is smaller than -10).

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Important commands in EViews

ar(1): Includes a single autoregressive lag

ar(2): Includes a second autoregressive lagNote, if you include only ar(2), EViews will not

include a first order autoregressive lag

ma(1): Includes a first order moving average term. This is not the same as forecasting using an average of recent values

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Selecting an appropriate time series model, concluded

Determine if trend/seasonality is important If it is, include it in your model

Estimate the model with necessary trend/seasonal components. Look at the correlogram of the residuals: From the equation dialogue box:

View => Residual Tests => Correlogram Q-statistics If ACs decay slowly with abrupt cutoff in PAC, this is indicative

of AR components. If the PAC doesn’t cutoff, you may need to include MA components as well.

Re-estimate the full model with trend/seasonality included with necessary ARMA components. You will likely have several models to choose from.

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Selecting an appropriate model, cont.

After you estimate each model, record SIC/AIC values

Use the SIC/AIC values to select an appropriate model.

Finally, investigate the final set of residuals. There should be no correlation in your residuals.Evidenced by individual correlation coefficients

within 95% confidence intervals about zero.Ljung-Box Q-statistics should be small with

probability values typically in excess of 0.05.

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Exponential smoothing

Eviews provides five options when you ask it, no tell it, to provide exponential smoothing:Single: (no seasonality/no trend)Double: (trend – value of =).Holt-Winters – No seasonal (Trend, and are not

equal, but are estimated in the data).Holt-Winters – Additive (Trend and Seasonality. The

seasonal component is estimated with an additive filter).

Holt-Winters - Multiplicative

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Moving average methods

Two-sided (for some arbitrary m):

One-sided:

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1

2m +1y t−i

i=−m

m

∑ , with m = 2 :

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5(y t +2 + y t +1 + y t + y t−1 + y t−2)

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1

m +1y t−i

i= 0

m

∑ , with m = 2 :

1

3(y t + y t−1 + y t−2)

To calculate a moving average:

Limit your sample to the period the average is being constructed under.

Create a variable that is simply an average of the variable you are looking for, for the restricted period.

For both the exponential smoothing model and moving average model, EViews will not give you the mean squared error. You will need to calculate it your self. You can do so in EViews or in Excel.

Breaks?

Uh oh? My data appears to have a break.

The developed time series methods assume the black box generating the data is constant.Not necessarily true:Learning curves may cause cost curves to

decreaseAcquisition of companies or new

technologies may alter sales/costs

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Dealing with breaks?

Solutions:Limit the sample to the post break periodSometimes taking logs and/or differencing can help

mitigate the effects of breaks/outliers. Include variables that help identify the breaks Model the breaks directly:

The most obvious way is to include a break in mean and/or a break in trend.

We should make sure that the included break is modeled in a sensible way A negative linear trend, for example, will imply the

data may eventually turn negative.

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Break in mean

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Break in trend

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Statistics useful in comparing the out of sample forecasting accuracy

Mean squared error: For an h-step extrapolation forecast:

Root mean squared error is the square root of this number.

Mean absolute error

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( ˆ Y T +1|T −YT +1)2 + ( ˆ Y T +2|T −YT +2)2 + ...+ ( ˆ Y T +h |T −YT +h )2

h

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| ˆ Y T +1|T −YT +1 | + | ˆ Y T +2|T −YT +2 | +...+ | ˆ Y T +h |T −YT +h |

h

In Eviews:

If you have a forecasted series, say xf, and an original series x, you can calculate the mean squared error as:

genr mse=@sum((x-xf)^2)/h

To calculate the moving average forecasts: Suppose you use the most recent four periods Limit your data set to include only the last four

observations A variable called “maf_4” is calculated by: genr maf_4=@mean(x)

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Lecture 8: Part 2

SHIFTS IN SUPPLY AND DEMANDUNDERSTANDING INDUSTRY CHANGES

– Summary of main points

• A market has a product, geographic, and time dimension. Define the market before using supply–demand analysis.

• Market demand describes buyer behavior; market supply describes seller behavior in a competitive market.

• If price changes, quantity demanded increases or decreases (represented by a movement along the demand curve).

• If a factor other than price (like income) changes, we say that demand curve increases or decreases (a shift of demand curve).

Lecture 8 – Summary (cont.)• Supply curves describe the behavior of sellers and tell

you how much will be sold at a given price.

• Market equilibrium is the price at which quantity supplied equals quantity demanded. If price is above the equilibrium price, there are too many sellers, forcing price down, and vice versa.

• Currency depreciation in a country increases demand for exports (supply to another country) and decreases demand for imports (demand for another country’s products).

• Prices are a primary way that market participants communicate with one another.

• Making a market is costly, and competition between market makers forces the bid–ask spread down to the costs of making a market. If the costs of making a market are large, then the equilibrium price may be better viewed as a spread rather than a single price.

Anecdote: Y2K and generator sales• From 1990-98, sales of portable generators

grew 2% yearly.

• In 1999, public anticipation of Y2K power outages increased demand for generators.• Walters, Rosenberg and Matthews invested to

increase capacity in anticipation of this demand growth – they vertically integrated their company to increase capacity and reduce variable costs.

• Demand grew as expected - Industry shipments increased by 87%. Prices also increased by an average of 21%.

Which industry or market?

• Every industry or market has a time, product, and geographic dimension.

• For example: The yearly market for portable generators in the U.S.

• Time: annual

• Product: portable generators

• Geography: US

• When analyzing a problem, or investment opportunity, it helps to first define the time, product and geographic dimensions of the market in question.

Shifts in the demand curve• Movement along the demand curve indicates the

“quantity demanded” increased.

• Shifts in demand curve can occur for multiple reasons

• Uncontrollable factor – affects demand and is out of a company’s control.

• Income, weather, interest rates, and prices of substitute and complementary products owned by other companies.

• Controllable factor – affects demand but can be controlled by a company

• Advertising, warranties, product quality, distribution speed, service quality, and prices of substitute or complementary products also owned by the company

Anecdote: Microsoft

• In the late 1970s, Microsoft developed DOS, an operating system to control IBM computers.

• The price for DOS depended on the price and availability of computers that could run it and the applications that ran under it as well as the price of DOS itself.

• To increase demand for DOS Microsoft:

• Licensed its operating system to other computer manufacturers

• Developed its own versions of complimentary products

• To affect the quantity demanded, they also kept the price of DOS low.

Demand increase• At a given price, more quantity demanded

Supply curves• Definition: Supply curves are functions that

relate the price of a product to the quantity supplied by sellers.

• Discussion: Why do supply curves slope upwards?

Market equilibrium

• Definition: Market equilibrium is the price at which quantity supplied equals quantity demanded.

• At the equilibrium price, there is no pressure for the price to change given the equality of quantity demanded and supplied.

Market equilibrium (cont.)• Proposition: In a competitive

equilibrium there are no unconsummated wealth-creating transactions.

Price Demand Supply$12 1 9$11 2 8$10 3 7$9 4 6$8 5 5$7 6 4$6 7 3$5 8 2$4 9 1

Using supply and demand• Supply and demand curves can be used to describe changes that occur at the industry level

Using supply and demand (cont.)• Discussion: “over the past decade, the price of

computers has fallen, while quantity has risen.” How? Why?

Prices convey information

• Prices are a primary way that market participants communicate with one another

• Buyers signal their willingness to pay, and sellers signal their willingness to sell with prices

• Price information especially important in financial markets