Price Forecasting Using Statistical Model
Data
1. Time series – data gathered sequentially in time;
correlation among observations usually exists.
2. Cross section – data for specific period of time consisting
of uncorrelated observation.
Model - a set of assumptions that summarizes a structure of
the system.
Purpose of Modeling
1. To understand the mechanism that generates the data
2. To forecast
3. To optimize the response of a variable of interest
Model Building Process
1. Planning
a. Define the problem
b. Determine an appropriate model
c. Define the variables
2. Development of the model
a. Collect data
b. Estimate the model
c. Statistical hypothesis
d. Diagnostic tests
3. Verification of the model
a. Forecasting ability
b. Usefulness of the model
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Models Used in Forecasting
1. Non-structured or time series model
2. Structured or econometric models
Structured or Causal or Fundamental or Econometric Models
- incorporates variables or factors that might influence
price, e.g. seasonal use patterns, seasonal supply
patterns, prices of related goods, market structure,
income.
- technique for investigating relationships among
variables causal models
- tool in research and planning
- applicable in social sciences, business, and economics
a. Variables
Y = dependent variable
X’s = independent variables
p = number of independent variables
n = number of observations
Example:
Pt = deflated season average cranberry price
Pt-1 = lagged deflated season average cranberry price
PCS = per capita supply of cranberry
PCI = per capita deflated disposable income
D = dummy equal to 0 for 1954-76 observations and PCS
for 1991-2002 observations
Have to match observations for all variables
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b. Linear Model
𝑷 = 𝜷𝒐 + 𝜷𝟏 𝑿𝟏 + 𝜷𝟐 𝑿𝟐 + ⋯ + 𝜷𝒑 𝑿𝒑 + 𝜺
Example:
𝑷𝒕 = 𝜷𝒐 + 𝜷𝟏 𝑷𝒕−𝟏 + 𝜷𝟐 𝑷𝑪𝑺 + 𝜷𝟑 𝑷𝑪𝑰 + 𝜷𝟒 𝑫 + 𝜺
c. Reasons for inclusion of the error term ε
i. to measure the error
ii. to account for other independent variables not in the
model
d. Regression assumptions
i. The average value of unknown variable is 0, μ = 0.
ii. The variance of error term is the same for all
observations, σ2 = c.
iii. The error terms among observations are
uncorrelated.
iv. The error term follow normal distribution.
v. The independent variables are not strongly
correlated.
e. Problems
i. To estimate the regression coefficients
ii. To verify the validity of assumptions
iii.To check the usefulness of the model
f. Estimation of regression coefficients
- least-squares method
minimizes the deviations of the observations from
the fitted line.
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g. Interpretation of coefficients
- in relation to theory
- effect on Y
- coefficients are slopes/partial elasticities
h. Significance of the effects of variables
- test whether independent variables affect dependent
variable.
t-test: large values support significance
p-value: small values imply difference
i. Regression tests
i. ANOVA – F-test
ii. Coefficient of determination – R2
iii. Diagnostic checking of residuals
iv. Heteroscedasticity – Goldfeld-Quandt test
v. Autocorrelation – Durbin Watson test
vi. Multicollinarity – ridge regression, deletion
vii. Use of dummy variables
j. Forecasting (prediction)
- based on the given levels of Xs, forecast the value of Y.
Non-Structured or Time-Series Models
- attempt to predict future using historical data.
- assume that what will happen in the future is a function
of what happened in the past.
1. Naïve Model
- assumes forecast for price in period t+1 is equivalent
to its immediate price in period t.
𝑷𝒕+𝟏 = 𝑷𝒕
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where
Pt+1 = forecast price in period t+1
Pt = actual price in period t
2. Moving Average
- assumes that price will stay fairly steady over time.
- smooths out sudden fluctuations in order to provide
stable estimates.
∑ 𝒑𝒓𝒊𝒄𝒆 𝒊𝒏 𝒑𝒓𝒆𝒗𝒊𝒐𝒖𝒔 𝒏 𝒑𝒆𝒓𝒊𝒐𝒅𝒔
𝑴𝒐𝒗𝒊𝒏𝒈 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 =
𝒏
Limitations
1. Increasing n does smooth out fluctuations better, but
it makes the method less sensitive to real changes in
the data.
2. It cannot pick up trends very well.
3. It requires extensive record keeping of past data when
n is large.
3. Exponential Smoothing
New forecast = last period’s forecast + α (last period’s
actual price – last period’s forecast)
𝑷𝒕+𝟏 = 𝑷𝒕 + 𝜶(𝑨𝑷𝒕 − 𝑷𝒕 )
𝑷𝒕+𝟏 = 𝜶𝑨𝑷𝒕 + (𝟏 − 𝜶)𝑷𝒕
where
Pt+1 = price forecast for period t+1
Pt = price forecast for period t
α = smoothing constant (0 ≤ α ≤1)
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4. Trend Projection
- fits a trend line to a series of historical data and then
projects the trend line into the future for medium- to
long-range forecasts
- mathematical trend equations:
Linear: 𝑷𝒕 = 𝜶 + 𝜷𝒕 + 𝜺
Exponential: 𝑷𝒕 = 𝜶𝒕𝜷
Quadratic: 𝑷𝒕 = 𝜶 + 𝜷𝒕 + 𝜹𝒕𝟐
5. Decomposition Method
Multiply the trend line forecast by the appropriate
seasonal index.
6. Random Walk Model
- states that price series do not exhibit predictive
patterns over time but can best be described with a
random walk.
7. Box-Jenkins Arima model
8. ARCH (Autoregressive Conditional Heteroskedastic) Model
9. GARCH (Generalized Autoregressive Conditional
Heteroskedastic) Model
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Measures of Forecast Accuracy
- to see how well one model works, or to compare
models, the forecasted values are compared with the
actual or observed values.
Forecast error = actual value – forecast value
1. Mean Absolute Deviation (MAD)
∑|𝒇𝒐𝒓𝒆𝒄𝒂𝒔𝒕 𝒆𝒓𝒓𝒐𝒓𝒔|
𝑴𝑨𝑫 =
𝒏
2. Mean Squared Error (MSE)
∑(𝒇𝒐𝒓𝒆𝒄𝒂𝒔𝒕 𝒆𝒓𝒓𝒐𝒓)𝟐
𝑴𝑺𝑬 =
𝒏
3. Mean Absolute Percent Error (MAPE)
- The average of the absolute values of the errors
expressed as percentage of the actual values
|𝒇𝒐𝒓𝒆𝒄𝒂𝒔𝒕 𝒆𝒓𝒓𝒐𝒓|
∑( ) ∗ 𝟏𝟎𝟎
𝒂𝒄𝒕𝒖𝒂𝒍 𝒗𝒂𝒍𝒖𝒆𝒔
𝑴𝑨𝑷𝑬 =
𝒏
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