Linear Models, GLS, and Seasonal Indicator Variables

Chapter 5: Lesson 1

Learning Outcomes

Explain the difference between stochastic and deterministic trends in time series

Describe deterministic trends as smooth, predictable changes over time
Define stochastic trends as random, unpredictable fluctuations
Explain the different treatment of stochastic and deterministic trends when forecasting

Fit linear regression models to time series data

Define a linear time series model
Explain why ordinary linear regression systematically underestimates of the standard error of parameter estimates when the error terms are autocorrelated
Apply generalized least squares GLS in R to estimate linear regression model parameters
Explain how to estimate the autocorrelation input for the GLS algorithm
Compare GLS and OLS standard error estimates to evaluate autocorrelation bias
Identify an appropriate function to model the trend in a given time series
Represent seasonal factors in a regression model using indicator variables
Fit a linear model for a simulated time series with linear time trend and $AR(p)$ error
Use acf and pacf to test for autocorrelation in the residuals
Estimate a seasonal indicator model using GLS
Forecast using a fitted GLS model with seasonal indicator variables

Apply differencing to nonstationary time series

Transform a non-stationary linear to a stationary process using differencing
State how to remove a polynomial trend of order $m$

Simulate time series

Simulate a time series with a linear time trend and a $AR(p)$ error

Preparation

Read Sections 5.1-5.5

Learning Journal Exchange (10 min)

Review another student’s journal
What would you add to your learning journal after reading another student’s?
What would you recommend the other student add to their learning journal?
Sign the Learning Journal review sheet for your peer

Small Group Activity: Linear Models (5 min)

Definition of Linear Models

Linear Model for a Time Series

A model for a time series $\{x_t : t = 1 \ldots n \}$ is linear if it can be written in the form \[ x_t = \alpha_0 + \alpha_1 u_{1,t} + \alpha_2 u_{2,t} + \alpha_3 u_{3,t} + \ldots + \alpha_m u_{m,t} + z_t \]

where $u_{i,t}$ represents the $i^{th}$ predictor variable observed at time $t$, $z_t$ is the value of the error time series at time $t$, the values $\{\alpha_0, \alpha_1, \alpha_2, \ldots, \alpha_m\}$ are model parameters, and $i = 1, 2, \ldots, n$ and $t = 1, 2, \ldots, m$.

The error terms $z_t$ have mean 0, but they do not need to follow any particular distribution or be independent.

With a partner, practice determining which models are linear and which are non-linear.

Check Your Understanding

Which of the following models are linear?
- Model 1: \[ x_t = \alpha_0 + \alpha_1 \sin(t^2-1) + \alpha_2 e^{t} + z_t \]
- Model 2: \[ x_t = \sqrt{ \alpha_0 + \alpha_1 t + z_t } \]
- Model 3: \[ x_t = \alpha_0 + \alpha_1 t^2 + \alpha_2 ( t \cdot \sqrt{t-3} ) + z_t \]
- Model 4: \[ x_t = \frac{ \alpha_0 + \alpha_1 t }{ 1 + \alpha_2 \sqrt{t+1} } + z_t \]
- Model 5: \[ x_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \alpha_3 t^3 + z_t \]
- Model 6: \[ x_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \alpha_1 \alpha_2 t^3 + z_t \]
Is there a way to transform the following non-linear models into a linear model? If so, apply the transformation.
- Model 7: \[ x_t = \sqrt{ \alpha_0 + \alpha_1 t + z_t } \]
- Model 8: \[ x_t = \sin( \alpha_1 + \alpha_2 t + z_t ) \]
- Model 9: \[ x_t = \alpha_0 \sin( \alpha_0 + \alpha_1 t + z_t ) \]
- Model 10: \[ x_t = \alpha_0 ~ e^{ \alpha_1 t + \alpha_2 t^2 + z_t } \]

Stationarity of Linear Models

Check Your Understanding

Complete the following quotes from page 93 of the textbook:

“Linear models for time series are non-stationary when they include functions of ______.”
______________________ can often transform a non-stationary series with a deterministic trend to a stationary series.

The book presents a time series that is modeled as a linear function of time plus white noise: \[ x_t = \alpha_0 + \alpha_1 t + z_t \] If we difference these values, we get: \[\begin{align*} \nabla x_t &= x_t - x_{t-1} \\ &= \left[ \alpha_0 + \alpha_1 t + z_t \right] - \left[ \alpha_0 + \alpha_1 (t-1) + z_t \right] \\ &= z_t - z_{t-1} + \alpha_1 \end{align*}\]

Given that the white noise process $\{ z_t \}$ is stationary, then $\{ \nabla x_t \}$ is stationary. Differencing is a powerful tool for making a time series stationary.

In Chapter 4 Lesson 2, we established that taking the difference of a non-stationary time series with a stochastic trend can convert it to a stationary time series. Later in the same lesson, we learn that a linear deterministic trend can be removed by taking the first difference. A quadratic deterministic trend can be removed by taking the second differences, and so on.

Note

In general, if there is an $m^{th}$ degree polynomial trend in a time series, we can remove it by taking $m$ differences.

Class Activity: Fitting Models (15 min)

Simulation

In Section 5.2.3, the textbook illustrates the time series \[ x_t = 50 + 3t + z_t \] where $\{ z_t \}$ is the $AR(1)$ process $z_t = 0.8 z_{t-1} + w_t$ and $\{ w_t \}$ is a Gaussian white noise process with $\sigma = 20$. The code below simulates this time series and creates the resulting time plot.

Show the code

set.seed(1)

dat <- tibble(w = rnorm(100, sd = 20)) |>
    mutate(
        Time = 1:n(),
         z = purrr::accumulate2(
            lag(w), w, 
            \(acc, nxt, w) 0.8 * acc + w,
            .init = 0)[-1],
          x = 50 + 3 * Time + z
            ) |>
    tsibble::as_tsibble(index = Time)
dat |> autoplot(.var = x) +
  theme_minimal()

Figure 1: Time plot of a simulated time series with a linear trend.

The advantage of simulation is that when we fit a model to the data, we can asses the fit…we know exactly what was simulated.

Model Fitted to the Simulated Data

When applying ordinary least-squares multiple regression, it is common to fit the model by minimizing the sum of squared errors: \[ \sum r_i^2 = \sum \left( x_t - \left[ \alpha_0 + \alpha_1 u_{1,t} + \alpha_2 u_{2,t} + \ldots + \alpha_m u_{m,t} \right] \right)^2 \] In R, we accomplish this using the lm function.

We assume that the simulated time series above follows the model \[ x_t = \alpha_0 + \alpha_1 t + z_t \]

Show the code

dat_lm <- dat |>
  model(lm = TSLM(x ~ Time))
params <- tidy(dat_lm) |> pull(estimate)
stderr <- tidy(dat_lm) |> pull(std.error)

This gives the fitted equation \[ \hat x_t = 58.551 + 3.063 t \]

The standard errors of these estimated parameters tend to be underestimated by the ordinary least squares method. To illustrate this, a simulation of n = 1000 realizations of the time series above was conducted.

The parameter estimates and their standard errors are summarized in the table below. The “Computed SE” is the standard error reported by the lm function in R. The “Simulated SE” is the standard deviation of the parameter estimated obtained in the 1000 simulated realizations of this time series.

Show the code

# This code simulates the data for this example
#
# parameter_est <- data.frame(alpha0 = numeric(), alpha1 = numeric())
# for(count in 1:n) {
#   dat <- tibble(w = rnorm(100, sd = 20)) |>
#       mutate(
#           Time = 1:n(),
#            z = purrr::accumulate2(
#               lag(w), w, 
#               \(acc, nxt, w) 0.8 * acc + w,
#               .init = 0)[-1],
#             x = 50 + 3 * Time + z
#               ) |>
#       tsibble::as_tsibble(index = Time)
#   
#   dat_lm <- dat |>
#   model(lm = TSLM(x ~ Time))
# 
#   parameters <- tidy(dat_lm) |> pull(estimate)
#   parameter_est <- parameter_est |>
#     bind_rows(data.frame(alpha0 = parameters[1], alpha1 = parameters[2])) 
# }
# 
# parameter_est |> 
#   rio::export("data/chapter_5_lesson_1_simulation.parquet")

parameter_est <- rio::import("https://byuistats.github.io/timeseries/data/chapter_5_lesson_1_simulation.parquet")


standard_errors <- parameter_est |>
  summarize(
    alpha0 = sd(alpha0), 
    alpha1 = sd(alpha1)
  )

Table 1: Autocorrelation function and partial autorcorrelation function of the residuals from the fitted linear model for the simulated data. Several realizations of the time series were simulated, and the standard deviation of the estimated parameters from the simulation is compared against the standard errors obtained by ordinary least squares regression.

Parameter	Estimate	Computed SE	Simulated SE
$\alpha_0$	58.551	4.88	17.798
$\alpha_1$	3.063	0.084	0.315

Warning

Note that the simulated standard errors are much larger than those obtained by the lm function in R. The standard errors reported by R will lead to unduly small confidence intervals. This illustrates the problem of applying the standard least-squares estimates to time series data.

After fitting a regression model, it is appropriate to review the relevant diagnostic plots. Here are the correlogram and partial correlogram of the residuals.

Show the code

acf_plot <- residuals(dat_lm) |> feasts::ACF() |> autoplot()

pacf_plot <- residuals(dat_lm) |> feasts::PACF() |> autoplot()

acf_plot | pacf_plot

Figure 2: Autocorrelation function and partial autorcorrelation function of the residuals from the fitted linear model for the simulated data.

Recall that in our simulation, the residuals were modeled by an $AR(1)$ process. So, it is not surprising that the residuals are correlated and that the partial autocorrelation function is only significant for the first lag.

Fitting a Regression Model to the Global Temperature Time Series

In Chapter 4 Lesson 2, we fit AR models to data representing the change in the Earth’s mean annual temperature from 1880 to 2023. These values represent the deviation of the mean global surface temperature from the long-term average from 1951 to 1980. (Source: NASA/GISS.) We will consider the portion of the time series beginning in 1970.

Show the code

temps_ts <- rio::import("https://byuistats.github.io/timeseries/data/global_temparature.csv") |>
  as_tsibble(index = year) |>
  filter(year >= 1970)

temps_ts |> autoplot(.vars = change) +
    labs(
      x = "Year",
      y = "Temperature Change (Celsius)",
      title = paste0("Change in Mean Annual Global Temperature (", min(temps_ts$year), "-", max(temps_ts$year), ")")
    ) +
    theme_minimal() +
    theme(
      plot.title = element_text(hjust = 0.5)
    ) + 
  geom_smooth(method='lm', formula= y~x)

Figure 3: Time plot of the mean annual temperature globally for select years.

Visually, it is easy to spot a positive trend in these data. Nevertheless, the ordinary least squares technique underestimates the standard error of the constant and slope term. This can lead to errant conclusions that a linear relationship is statistically significant when it is not.

Show the code

global <- tibble(x = scan("https://byuistats.github.io/timeseries/data/global.dat")) |>
    mutate(
        date = seq(
            ymd("1856-01-01"),
            by = "1 months",
            length.out = n()),
        year = year(date),
        year_month = tsibble::yearmonth(date),
        stats_time = year + c(0:11)/12)

global_ts <- global |>
    as_tsibble(index = year_month)

temp_tidy <- global_ts |> filter(year >= 1970)
temp_lm <- temp_tidy |>
  model(lm = TSLM(x ~ stats_time ))
# tidy(temp_lm) |> pull(estimate)

tidy(temp_lm) |>
  mutate(
    lower = estimate + qnorm(0.025) * std.error,
    upper = estimate + qnorm(0.975) * std.error
  ) |>
  display_table()

Table 2: Parameter estimates for the fitted global temperature time series.

.model	term	estimate	std.error	statistic	p.value	lower	upper
lm	(Intercept)	-34.920409	1.164899	-29.97721	0	-37.2035680	-32.6372500
lm	stats_time	0.017654	0.000586	30.12786	0	0.0165055	0.0188025

The 95% confidence interval for the slope does not contain zero. If the errors were independent, this would be conclusive evidence that there is a significant linear relationship between the year and the global temperature.

Note that there is significant positive autocorrelation in the residual series for short lags. This implies that the standard errors will be underestimated, and the confidence interval is inappropriately narrow.

Here are the values of the autocorrelation function of the residuals:

Table 3: Autocorrelation function of the residuals for the global temperature time series.

1M	2M	3M	4M	5M	6M	7M	8M	9M	10M
0.706	0.638	0.5	0.447	0.375	0.307	0.254	0.206	0.12	0.078

The autocorrelation for $k=1$ is 0.706. We will use this value when we apply the Generalized Least Squares method in the next section.

To get an intuitive idea of why ordinary least squares underestimates the standard errors of the estimates, the positive autocorrelation of adjacent observations leads to a series of data that have an effective record length that is shorter than the number of observations. This happens because similar observations tend to be clumped together, and the overall variation is thereby understated.

Applying Generalized Least Squares, GLS

The autocorrelation in the data make ordinary least squares estimation inappropriate. What caped superhero comes to our rescue? None other than Captain GLS – the indominable Generalized Least Squares algorithm!

This fitting procedure handles the autocorrelation by maximizing the likelihood of the data, taking into account the autocorrelation in the residuals. This leads to much more appropriate standard errors for the parameter estimates. We will pass the value of the acf at lag $k=1$ into the regression function.

# Load additional packages
pacman::p_load(tidymodels, multilevelmod,
  nlme, broom.mixed)

temp_spec <- linear_reg() |>
  set_engine("gls", correlation = nlme::corAR1(0.706))

temp_gls <- temp_spec |>
  fit(x ~ stats_time, data = global_ts)

tidy(temp_gls) |>
  mutate(
    lower = estimate + qnorm(0.025) * std.error,
    upper = estimate + qnorm(0.975) * std.error
  )

Table 4: Parameter estimates for the Generalized Least Squares estimates of the fit for the global temperature time series.

term	estimate	std.error	statistic	p.value	lower	upper
(Intercept)	-9.1361773	0.4547490	-20.09059	0	-10.0274689	-8.2448856
stats_time	0.0046599	0.0002354	19.79209	0	0.0041985	0.0051214

As we observed visually, there is evidence to suggest that there is a linear trend in the data.

Class Activity: Linear Models with Seasonal Variables (10 min)

Some time series involve seasonal variables. In this section, we will address one way to include seasonality in a regression analysis of a time series. In the next lesson, we will explore another method.

Additive Seasonal Indicator Variables

Recall the time series representing the monthly relative number of Google searches for the term “chocolate.” Here is a time plot of the data from 2021 to 2023. Month 1 is January 2021, month 13 is January 2022, and month 25 is January 2023.

Figure 4: Time plot of a portion of the chocolate search data.

We fit a regression line to these data.

Figure 5: The regression line is superimposed on the chocolate search time series.

Now, we draw a line that is parallel to the regression line (has the same slope) but has a Y-intercept of 0.

Figure 6: A line parallel to the regression line that passes through the origin is added to the chocolate search time plot.

For each month, we find the average amount that the relative number of google seachers that month deviates from the orange line (that is parallel to the regression line and passes through the origin). So, the length of the green line is the same for every January, etc.

Figure 7: Representation of a linear model with seasonal indicator variables for the chocolate search time series data.

When the bottom of the green arrow is on the orange line, the top of the green arrow is the estimate of the value of the time series for that month.

We will create a linear model that includes a constant term for each month. This constant monthly term is called a seasonal indicator variable. This name is derived from the fact that each variable indicates (either as 1 or 0) whether a given month is represented. For example, one of the seasonal indicator variables will represent January. It will be equal to 1 for any value of $t$ representing an observation drawn in January and 0 otherwise. Indicator variables are also called dummy varaibles.

This additive model with seasonal indicator variables can be perceived similarly to other additive models with a seasonal component:

\[ x_t = m_t + s_t + z_t \] where \[ s_t = \begin{cases} \beta_1, & t ~\text{falls in season}~ 1 \\ \beta_2, & t ~\text{falls in season}~ 2 \\ ⋮~~~~ & ~~~~~~~~~~~~⋮ \\ \beta_s, & t ~\text{falls in season}~ s \end{cases} \] and $s$ is the number of seasons in one cycle/period, and $n$ is the number of observations, so $t = 1, 2, \ldots, n$ and $i = 1, 2, \ldots, s$, and $z_t$ is the residual error series, which can be autocorrelated.

It is important to note that $m_t$ does not need to be a constant. It can be a linear trend: \[ m_t = \alpha_0 + \alpha_1 t \] or quadratic: \[ m_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 \] a polynomial of degree $p$: \[ m_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \cdots + \alpha_p t^p \] or any other function of $t$.

If $s_t$ has the same value for all corresponding seasons, then we can write the model as: \[ x_t = m_t + \beta_{1 + [(t-1) \mod s]} + z_t \]

Putting this all together, if we have a time series that has a linear trend and monthly observations where $t=1$ corresponds to January, then the model becomes: \[\begin{align*} x_t &= ~~~~ \alpha_1 t + s_t + z_t \\ &= \begin{cases} \alpha_1 t + \beta_1 + z_t, & t = 1, 13, 25, \ldots ~~~~ ~~(January) \\ \alpha_1 t + \beta_2 + z_t, & t = 2, 14, 26, \ldots ~~~~ ~~(February) \\ ~~~~~~~~⋮ & ~~~~~~~~~~~~⋮ \\ \alpha_1 t + \beta_{12} + z_t, & t = 12, 24, 36, \ldots ~~~~ (December) \end{cases} \end{align*}\]

This is the model illustrated in Figure 7. The orange line represents the term $\alpha_1 t$ and the green arrows represent the values of $\beta_1, ~ \beta_2, ~ \ldots, ~ \beta_{12}$.

The folded chunk of code below fits the model to the data and computes the estimated parameter values.

Show the code

chocolate_month <- rio::import("https://byuistats.github.io/timeseries/data/chocolate.csv") |>
  mutate(
    dates = yearmonth(ym(Month)),
    month = month(dates),
    year = year(dates),
    stats_time = year + (month - 1) / 12,
    month_seq = 1:n()
  ) |>
  mutate(month = factor(month)) |>
  as_tsibble(index = dates)

# Fit regression model
chocolate_lm <- chocolate_month |>
  model(TSLM(chocolate ~ 0 + stats_time + month))

# Estimated parameter values
param_est <- chocolate_lm |>
  tidy() |>
  pull(estimate)

The estimated value of $\alpha_1$ is 1.1316416. The estimated values for the $\beta_i$ parameters are:

Table 5: Estimated values of $eta_i$ for the fitted model for chocolate search time series.

Jan	Feb	Mar	Apr	May	Jun
-2227.91	-2219.305	-2233.149	-2233.443	-2234.838	-2237.932

Jul	Aug	Sep	Oct	Nov	Dec
-2236.476	-2237.37	-2237.515	-2232.659	-2223.003	-2200.098

Now, we compute forecasted (future) values for the relative number of Google searches for the word “Chocolate.”

Show the code

num_years_to_forecast <- 5

new_dat <- chocolate_month |>
  as_tibble() |>
  tail(num_years_to_forecast * 12) |>
  dplyr::select(stats_time, month) |>
  mutate(
    stats_time = stats_time + num_years_to_forecast,
    alpha = tidy(chocolate_lm) |> slice(1) |> pull(estimate),
    beta = rep(tidy(chocolate_lm) |> slice(2:13) |> pull(estimate), num_years_to_forecast)
  )

chocolate_forecast <- chocolate_lm |>
  forecast(new_data=as_tsibble(new_dat, index = stats_time)) 

chocolate_forecast |> 
  autoplot(chocolate_month, level = 95) +
  labs(
      x = "Month",
      y = "Relative Count of Google Searches",
      title = paste0("Google Searches for 'Chocolate' (", min(chocolate_month$year), "-", max(chocolate_month$year), ")"),
      subtitle = paste0(num_years_to_forecast, "-Year Forecast Based on a Linear Model with Seasonal Indicator Variables")
    ) +
    theme_minimal() +
    theme(
      plot.title = element_text(hjust = 0.5),
      plot.subtitle = element_text(hjust = 0.5)
    )

Figure 8: Forecasted values of the relative number of Google searches for the word ‘chocolate.’

Show the code

new_dat |>
  mutate(estimate = alpha * stats_time + beta) |>
  display_partial_table(nrow_head = 4, nrow_tail = 3, decimals = 3)

Table 6: Select forecasted values of the relative count of Google searches for ‘chocolate’.

stats_time	month	alpha	beta	estimate
2024	1	1.132	-2227.91	62.532
2024.083	2	1.132	-2219.305	71.232
2024.167	3	1.132	-2233.149	57.482
2024.25	4	1.132	-2233.443	57.282
⋮	⋮	⋮	⋮	⋮
2028.75	10	1.132	-2232.659	63.159
2028.833	11	1.132	-2223.003	72.909
2028.917	12	1.132	-2200.098	95.909

Table 6 gives a few of the forecasted values illustrated in Figure 8.

Small Group Activity: Application to Retail Sales Data (15 min)

Recall the monthly U.S. retail sales data. We will consider the total retail sales for women’s clothing for 2004-2006. You can use the code below to download the data into a tsibble.

# Read in Women's Clothing Retail Sales data
retail_ts <- rio::import("https://byuistats.github.io/timeseries/data/retail_by_business_type.parquet") |>
  filter(naics == 44812) |>
  filter(month >= yearmonth(my("Jan 2004")) & month <= yearmonth(my("Dec 2006"))) |>
  mutate(month_seq = 1:n()) |>
  mutate(year = year(month)) |>
  mutate(month_num = month(month)) |>
  as_tsibble(index = month)

Figure 9: Time plot of the aggregated monthly sales for women’s clothing stores in the United States.

We define the model as : \[\begin{align*} x_t &= ~~~~ \alpha_1 t + s_t + z_t \\ &= \begin{cases} \alpha_1 t + \beta_1 + z_t, & t = 1, 13, 25 ~~~~ ~~(January) \\ \alpha_1 t + \beta_2 + z_t, & t = 2, 14, 26 ~~~~ ~~(February) \\ ~~~~~~~~⋮ & ~~~~~~~~~~~~⋮ \\ \alpha_1 t + \beta_{12} + z_t, & t = 12, 24, 36 ~~~~ (December) \end{cases} \end{align*}\]

When we regress the women’s clothing retail sales in millions, $x_t$, on the month number, $t$, the estimated simple linear regression equation is: \[ \hat x_t = 2635 + 24 ~ t \] where $t = 1, 2, 3, \ldots, 35, 36$.

Check Your Understanding

Use the estimated regression equation \[ \hat x_t = 2635 + 24 ~ t \] to do the following:

Find the equation for the line parallel to the regression line that passes through the origin.
Compute the deviation of each observed point from the line you just obtained.
Compute the value of $\hat \beta_i$ for each month, where $i = 1, 2, \ldots, 12$.
Compute the estimate of the time series for $t = 1, 2, \ldots, 36$.
Predict the value of the time series for the next six months.

Table 7: Table used to compute the estimate of the women’s clothing retail sales time series using a linear model with seasonal indicator variables.

$$Date$$	$$t$$	$$x_t$$	$$\hat \alpha_1 t$$	$$x_t - \hat \alpha_1 t$$	$$\hat\beta_i$$	$$\hat x_t$$
Jan 2004	1	2267	24	2243	2062.33	2086.33
Feb 2004	2	2402	48	2354
Mar 2004	3	2915
Apr 2004	4	3023
May 2004	5	3005
Jun 2004	6	2777	144	2633	2569.67	2713.67
Jul 2004	7	2595	168	2427	2297	2465
Aug 2004	8	2644	192	2452	2345	2537
Sep 2004	9	2774	216	2558	2463	2679
Oct 2004	10	2990	240	2750	2574.33	2814.33
Nov 2004	11	3090	264	2826	2730	2994
Dec 2004	12	4472	288	4184	4173	4461
Jan 2005	13	2338	312	2026	2062.33	2374.33
Feb 2005	14	2453	336	2117
Mar 2005	15	3105
Apr 2005	16	3177
May 2005	17	3164
Jun 2005	18	3045	432	2613	2569.67	3001.67
Jul 2005	19	2741	456	2285	2297	2753
Aug 2005	20	2871	480	2391	2345	2825
Sep 2005	21	2963	504	2459	2463	2967
Oct 2005	22	3120	528	2592	2574.33	3102.33
Nov 2005	23	3342	552	2790	2730	3282
Dec 2005	24	4756	576	4180	4173	4749
Jan 2006	25	2518	600	1918	2062.33	2662.33
Feb 2006	26	2518	624	1894
Mar 2006	27	3194
Apr 2006	28	3379
May 2006	29	3340
Jun 2006	30	3183	720	2463	2569.67	3289.67
Jul 2006	31	2923	744	2179	2297	3041
Aug 2006	32	2960	768	2192	2345	3113
Sep 2006	33	3164	792	2372	2463	3255
Oct 2006	34	3197	816	2381	2574.33	3390.33
Nov 2006	35	3414	840	2574	2730	3570
Dec 2006	36	5019	864	4155	4173	5037
Jan 2007	37	—	888	—	2062.33	2950.33
Feb 2007	38	—	912	—
Mar 2007	39	—		—
Apr 2007	40	—		—
May 2007	41	—		—
Jun 2007	42	—	1008	—	2569.67	3577.67

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Retail Sales