Time Trends
• Simplest time trend is a linear trend
• Examine National Population data set.
• How well does a linear model work?
• Did you examine the residuals plots?
Time Trends
• Examine National Population data set.
• Make a prediction of U.S. Population in year
2011. Comment on your prediction.
Time Trends
• Recall:
Difference between the 95% CI and the 95% PI
• Confidence interval of the prediction:
Represents a range that the mean response is
likely to fall given specified settings of the
predictors.
• Prediction Interval: Represents a range that a
single new observation is likely to fall given
specified settings of the predictors.
Time Trends
• Simplest time trend is a linear trend
• Examine World Population data set.
• Notice there is not data for each year! How
can you make an appropriate time series
plot?
• How well does a linear model work?
• Did you examine the residuals plots?
Time Trends
• Examine World Population data set.
• Is there a model that might work better than
a linear model?
• How can you use linear regression with a
non-linear model?
Time Trends
• Using Time Series Trend Analysis in
Minitab
• Examine the U.S. population data set again.
Time Trends – Economics Example
• Open HSEINV data set
• invpc is real per capita housing investment in
thousands of dollars
• price is housing price index
Do invpc and price exhibit linear trends through
time?
Are invpc and price linearly related to each
other?
Time Trends – Economics Example
Book author fits this constant elasticity
model:
log(𝑖𝑛𝑣𝑝𝑐) = 𝛽𝑜 + 𝛽1 log(𝑝𝑟𝑖𝑐𝑒) + 𝑢
What do you think of this model?
How is invpc affected by price?
Time Trends – Economics Example
Author goes on to argue that both invpc and
price have upward time trends and the
model we just fit does not account for this.
Now, fit this model:
log 𝑖𝑛𝑣𝑝𝑐 = 𝛽𝑜 + 𝛽1 log 𝑝𝑟𝑖𝑐𝑒 + 𝛽2 𝑡 + 𝑢
Time Trends – Economics Example
Are your conclusions regarding How invpc is
affected by price different for the two
models?
Time Trends – Economics Example
Are your conclusions regarding How invpc is
affected by price different for the two
models?
From the second analysis, real per capita
housing investments are not influenced at all
by price once time is accounted for.
Time Trends – Economics Example
Are your conclusions regarding How invpc is
affected by price different for the two
models?
The first analysis showed a spruious
relationship between invpc and price due to
the fact that both variables are trending
upward over time.
Time Trends – Fertility Rate
Example Again!
We fit this model:
𝑔𝑓𝑟𝑡 = 𝛽𝑜 + 𝛽1 𝑝𝑒𝑡 + 𝛽2 𝑤𝑤2𝑡 + 𝛽3 𝑝𝑖𝑙𝑙𝑡 + 𝑢𝑡
It was a decent model.
Time Trends – Fertility Rate
Example Again!
Now, fit this model:
𝑔𝑓𝑟𝑡 = 𝛽𝑜 + 𝛽1 𝑝𝑒𝑡 + 𝛽2 𝑤𝑤2𝑡 + 𝛽3 𝑝𝑖𝑙𝑙𝑡 +𝛽4 𝑡 + 𝑢𝑡
Comment…
Time Trends – Fertility Rate
Example Again!
But wait, gfr does not follow a strictly linear trend
through time:
Time Trends – Fertility Rate
Example Again!
Why not just add a squared time term to the
model too. This is now a quadratic model in time:
𝑔𝑓𝑟𝑡
= 𝛽𝑜 + 𝛽1 𝑝𝑒𝑡 + 𝛽2 𝑤𝑤2𝑡 + 𝛽3 𝑝𝑖𝑙𝑙𝑡 +𝛽4 𝑡 + 𝛽5 𝑡 2 + 𝑢𝑡
Comment…
Time Trends – Fertility Rate
Example Again!
But wait, gfr does not follow a quadratic trend
through time:
Time Trends – Fertility Rate
Example Again!
Why not just add a squared and a cubed time term
to the model. This is now a cubic model in time:
𝑔𝑓𝑟𝑡
= 𝛽𝑜 + 𝛽1 𝑝𝑒𝑡 + 𝛽2 𝑤𝑤2𝑡
+ 𝛽3 𝑝𝑖𝑙𝑙𝑡 +𝛽4 𝑡 + 𝛽5 𝑡 2 + 𝛽6 𝑡 3 + 𝑢𝑡
Warning – this is starting to border on “curve
fitting”
Time Trends – Fertility Rate
Example Again!
Adding more polynomial terms in t allows us to
model any time series pretty well.
But,
• Model gets overly complicated
• We are just playing “connect-the-dots” and
missing broad trends in the data
• This offers little help in finding important
explanatory variables
Time Trends – Cheese!
Open the CHEESE data set which contains U.S.
production of blue and gorgonzola cheeses over
many years.
Is there a linear trend?
Fit this model: 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛𝑡 = 𝛽𝑜 +𝛽1 𝑡 + 𝑢𝑡
Time Trends – Cheese!
How is this model: 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛𝑡 = 𝛽𝑜 +𝛽1 𝑡 + 𝑢𝑡 ?
What did the model tell you about explanatory
variables that affect blue and gorgonzola cheese
production?
Stationary Time Series
Definition: A stationary time series process is one
in which the probability distribution(s) that
generate the time series are stable over time.
In other words, if we take any consecutive
collection of random variables in the series and
shift it ahead or back h time periods, the
probability distribution(s) remain unchanged.
Stationary Time Series – Example
Pharmaceutical Product Sales
Stationary Time Series – Example
How do we know Pharmaceutical Product Sales is
a stationary process?
Things to examine:
• No time effect
• Lag scatter plots
• Sample Autocorrelation Function (ACF)
Stationary Time Series – Example
Lets examine time effect in the Pharmaceutical
Product Sales data
How do we do this?
Lets examine time effect in the Pharmaceutical
Product Sales data
How do we do this? Regress the data against time
(or maybe time and time squared)
Pharmaceutical Product Sales regressed against time
(week)
Sales, in Thousands = 10368 + 0.184 Week
Predictor Coef SE Coef
T P
Constant 10368.0 40.1 258.58 0.000
Week
0.1844 0.5751 0.32 0.749
S = 218.244 R-Sq = 0.1% R-Sq(adj) = 0.0%
NO TIME EFFECT
Pharmaceutical Product Sales regressed against time (week)
and time squared
Sales, in Thousands = 10405 - 1.62 Week + 0.0149 Week
Squared
Predictor
Coef SE Coef
T P
Constant 10404.6 60.9 170.93 0.000
Week
-1.618 2.322 -0.70 0.487
Week Squared 0.01490 0.01859 0.80 0.425
S = 218.576 R-Sq = 0.6% R-Sq(adj) = 0.0%
NO TIME EFFECT
Pharmaceutical Product Sales has no time effect.
What does no time effect imply – constant mean
Estimate the constant mean of the Pharmaceutical
Product Sales data.
Stationary Time Series – Example
Lets examine lag scatter plots with the
Pharmaceutical Product Sales data.
Make new lag plus 1 variable in Minitab
Make scatter plot of data vs. lag plus 1
Scatter plot of data vs. lag plus 1
What does this graph imply?
Stationary Time Series – Example
Can explore other lags
Make new lag plus 2 variable in Minitab
Make scatter plot of data vs. lag plus 2
Scatter plot of data vs. lag plus 2
What does this graph imply?
Stationary Time Series – Example
Lets examine the Sample Autocorrelation Function
What is an autocorrelation function?
What is an autocorrelation function?
Autocorrelation coefficient at lag k is:
Cov 𝑦𝑡 , 𝑦𝑡+𝑘
𝜌𝑘 =
Var 𝑦𝑡
The collection of 𝜌𝑘 , 𝑘 = 1, 2, ⋯ is called the
autocorrelation function (ACF).
What is an autocorrelation function?
Autocorrelation coefficient at lag k is:
Cov 𝑦𝑡 , 𝑦𝑡+𝑘
𝜌𝑘 =
Var 𝑦𝑡
What is a variance? Var 𝑦𝑡 = Standard
What is an autocorrelation function?
Autocorrelation coefficient at lag k is:
Cov 𝑦𝑡 , 𝑦𝑡+𝑘
𝜌𝑘 =
Var 𝑦𝑡
What is a covariance?
Cov 𝑦𝑡 , 𝑦𝑡+𝑘 = 𝐸 𝑦𝑡 − 𝜇 𝑦𝑡+1 − 𝜇
What is a covariance?
covariance is a measure of how much two random
variables change together.
What is a covariance?
If the greater values of one variable mainly
correspond with the greater values of the other
variable, the covariance is a positive number.
If the greater values of one variable mainly
correspond to the smaller values of the other, the
covariance is negative.
The sign of the covariance therefore shows the
tendency in the linear relationship between the
variables.
Stationary Time Series – Example
Lets examine the Sample Autocorrelation Function
Minitab will estimate the Autocorrelation
Function from a set of time series data – the
Sample Autocorrelation Function
Stationary Time Series – Example
For the pharmaceutical product sales data:
Lag
ACF T LBQ
1 0.112486 1.23 1.56
2 0.012543 0.14 1.58
3 -0.223825 -2.42 7.84
4 -0.193314 -2.00 12.56
5 -0.113943 -1.14 14.21
6 0.014538 0.14 14.24
7 0.078927 0.78 15.05
8 0.045591 0.45 15.32
9 0.000628 0.01 15.32
Stationary Time Series – Example
The graph is usually more useful
Stationary Time Series – Example
Open Stationary Time Series Example data set and
explore the following
• Time effect present?
• What do lag scatter plots tell us?
• What does ACF tell us?
Is this data set a stationary time series?
Stationary Time Series – Example
Open Cheese data set and explore the following
• Time effect present?
• What do lag scatter plots tell us?
• What does ACF tell us?
Is this data set a stationary time series?