Description Usage Arguments Details See Also Examples
Interface to lm.wfit
for fitting dynamic linear models
and time series regression relationships.
1 2 3 
formula 
a 
data 
an optional 
subset 
an optional vector specifying a subset of observations to be used in the fitting process. 
weights 
an optional vector of weights to be used
in the fitting process. If specified, weighted least squares is used
with weights 
na.action 
a function which indicates what should happen
when the data contain 
method 
the method to be used; for fitting, currently only

model, x, y, qr 
logicals. If 
singular.ok 
logical. If 
contrasts 
an optional list. See the 
offset 
this can be used to specify an a priori
known component to be included in the linear predictor
during fitting. An 
start 
start of the time period which should be used for fitting the model. 
end 
end of the time period which should be used for fitting the model. 
... 
additional arguments to be passed to the low level regression fitting functions. 
The interface and internals of dynlm
are very similar to lm
,
but currently dynlm
offers three advantages over the direct use of
lm
: 1. extended formula processing, 2. preservation of time series
attributes, 3. instrumental variables regression (via twostage least squares).
For specifying the formula
of the model to be fitted, there are
additional functions available which allow for convenient specification
of dynamics (via d()
and L()
) or linear/cyclical patterns
(via trend()
, season()
, and harmon()
).
All new formula functions require that their arguments are time
series objects (i.e., "ts"
or "zoo"
).
Dynamic models: An example would be d(y) ~ L(y, 2)
, where
d(x, k)
is diff(x, lag = k)
and L(x, k)
is
lag(x, lag = k)
, note the difference in sign. The default
for k
is in both cases 1
. For L()
, it
can also be vectorvalued, e.g., y ~ L(y, 1:4)
.
Trends: y ~ trend(y)
specifies a linear time trend where
(1:n)/freq
is used by default as the regressor. n
is the
number of observations and freq
is the frequency of the series
(if any, otherwise freq = 1
). Alternatively, trend(y, scale = FALSE)
would employ 1:n
and time(y)
would employ the original time index.
Seasonal/cyclical patterns: Seasonal patterns can be specified
via season(x, ref = NULL)
and harmonic patterns via
harmon(x, order = 1)
.
season(x, ref = NULL)
creates a factor with levels for each cycle of the season. Using
the ref
argument, the reference level can be changed from the default
first level to any other. harmon(x, order = 1)
creates a matrix of
regressors corresponding to cos(2 * o * pi * time(x))
and
sin(2 * o * pi * time(x))
where o
is chosen from 1:order
.
See below for examples and M1Germany
for a more elaborate application.
Furthermore, a nuisance when working with lm
is that it offers only limited
support for time series data, hence a major aim of dynlm
is to preserve
time series properties of the data. Explicit support is currently available
for "ts"
and "zoo"
series. Internally, the data is kept as a "zoo"
series and coerced back to "ts"
if the original dependent variable was of
that class (and no internal NA
s were created by the na.action
).
To specify a set of instruments, formulas of type y ~ x1 + x2  z1 + z2
can be used where z1
and z2
represent the instruments. Again,
the extended formula processing described above can be employed for all variables
in the model.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64  ###########################
## Dynamic Linear Models ##
###########################
## multiplicative SARIMA(1,0,0)(1,0,0)_12 model fitted
## to UK seatbelt data
data("UKDriverDeaths", package = "datasets")
uk < log10(UKDriverDeaths)
dfm < dynlm(uk ~ L(uk, 1) + L(uk, 12))
dfm
## explicitly set start and end
dfm < dynlm(uk ~ L(uk, 1) + L(uk, 12), start = c(1975, 1), end = c(1982, 12))
dfm
## remove lag 12
dfm0 < update(dfm, . ~ .  L(uk, 12))
anova(dfm0, dfm)
## add season term
dfm1 < dynlm(uk ~ 1, start = c(1975, 1), end = c(1982, 12))
dfm2 < dynlm(uk ~ season(uk), start = c(1975, 1), end = c(1982, 12))
anova(dfm1, dfm2)
plot(uk)
lines(fitted(dfm0), col = 2)
lines(fitted(dfm2), col = 4)
## regression on multiple lags in a single L() call
dfm3 < dynlm(uk ~ L(uk, c(1, 11, 12)), start = c(1975, 1), end = c(1982, 12))
anova(dfm, dfm3)
## Examples 7.11/7.12 from Greene (1993)
data("USDistLag", package = "lmtest")
dfm1 < dynlm(consumption ~ gnp + L(consumption), data = USDistLag)
dfm2 < dynlm(consumption ~ gnp + L(gnp), data = USDistLag)
plot(USDistLag[, "consumption"])
lines(fitted(dfm1), col = 2)
lines(fitted(dfm2), col = 4)
if(require("lmtest")) encomptest(dfm1, dfm2)
###############################
## Time Series Decomposition ##
###############################
## airline data
data("AirPassengers", package = "datasets")
ap < log(AirPassengers)
ap_fm < dynlm(ap ~ trend(ap) + season(ap))
summary(ap_fm)
## Alternative time trend specifications:
## time(ap) 1949 + (0, 1, ..., 143)/12
## trend(ap) (1, 2, ..., 144)/12
## trend(ap, scale = FALSE) (1, 2, ..., 144)
## Exhibit 3.5/3.6 from Cryer & Chan (2008)
if(require("TSA")) {
data("tempdub", package = "TSA")
td_lm < dynlm(tempdub ~ harmon(tempdub))
summary(td_lm)
plot(tempdub, type = "p")
lines(fitted(td_lm), col = 2)
}

Loading required package: zoo
Attaching package: 'zoo'
The following objects are masked from 'package:base':
as.Date, as.Date.numeric
Time series regression with "ts" data:
Start = 1970(1), End = 1984(12)
Call:
dynlm(formula = uk ~ L(uk, 1) + L(uk, 12))
Coefficients:
(Intercept) L(uk, 1) L(uk, 12)
0.1826 0.4310 0.5112
Time series regression with "ts" data:
Start = 1975(1), End = 1982(12)
Call:
dynlm(formula = uk ~ L(uk, 1) + L(uk, 12), start = c(1975, 1),
end = c(1982, 12))
Coefficients:
(Intercept) L(uk, 1) L(uk, 12)
0.5418 0.2072 0.6228
Analysis of Variance Table
Model 1: uk ~ L(uk, 1)
Model 2: uk ~ L(uk, 1) + L(uk, 12)
Res.Df RSS Df Sum of Sq F Pr(>F)
1 94 0.24019
2 93 0.13598 1 0.10421 71.272 4.007e13 ***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Analysis of Variance Table
Model 1: uk ~ 1
Model 2: uk ~ season(uk)
Res.Df RSS Df Sum of Sq F Pr(>F)
1 95 0.32151
2 84 0.07277 11 0.24874 26.102 < 2.2e16 ***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Analysis of Variance Table
Model 1: uk ~ L(uk, 1) + L(uk, 12)
Model 2: uk ~ L(uk, c(1, 11, 12))
Res.Df RSS Df Sum of Sq F Pr(>F)
1 93 0.13598
2 92 0.13253 1 0.0034481 2.3936 0.1253
Loading required package: lmtest
Encompassing test
Model 1: consumption ~ gnp + L(consumption)
Model 2: consumption ~ gnp + L(gnp)
Model E: consumption ~ gnp + L(consumption) + L(gnp)
Res.Df Df F Pr(>F)
M1 vs. ME 15 1 12.569 0.0029371 **
M2 vs. ME 15 1 27.093 0.0001067 ***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Time series regression with "ts" data:
Start = 1949(1), End = 1960(12)
Call:
dynlm(formula = ap ~ trend(ap) + season(ap))
Residuals:
Min 1Q Median 3Q Max
0.156370 0.041016 0.003677 0.044069 0.132324
Coefficients:
Estimate Std. Error t value Pr(>t)
(Intercept) 4.726780 0.018894 250.180 < 2e16 ***
trend(ap) 0.120826 0.001432 84.399 < 2e16 ***
season(ap)Feb 0.022055 0.024211 0.911 0.36400
season(ap)Mar 0.108172 0.024212 4.468 1.69e05 ***
season(ap)Apr 0.076903 0.024213 3.176 0.00186 **
season(ap)May 0.074531 0.024215 3.078 0.00254 **
season(ap)Jun 0.196677 0.024218 8.121 2.98e13 ***
season(ap)Jul 0.300619 0.024221 12.411 < 2e16 ***
season(ap)Aug 0.291324 0.024225 12.026 < 2e16 ***
season(ap)Sep 0.146690 0.024229 6.054 1.39e08 ***
season(ap)Oct 0.008532 0.024234 0.352 0.72537
season(ap)Nov 0.135186 0.024240 5.577 1.34e07 ***
season(ap)Dec 0.021321 0.024246 0.879 0.38082

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0593 on 131 degrees of freedom
Multiple Rsquared: 0.9835, Adjusted Rsquared: 0.982
Fstatistic: 649.4 on 12 and 131 DF, pvalue: < 2.2e16
Loading required package: TSA
Attaching package: 'TSA'
The following objects are masked from 'package:stats':
acf, arima
The following object is masked from 'package:utils':
tar
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