This has been a difficult year for the Canadian dollar. Ever since oil prices started falling in mid 2014, the Canadian dollar has traded lower. Currently, in this low interest rate environment, oil prices seem to be the main driver of the Canadian dollar. From an international investment perspective, Canada is seen as consisting of safe banks and companies specializing in extracting natural resources. As a result, our currency and stock market both tend to follow commodity prices. Right now, commodity prices are depressed (especially oil) and this has taken the Canadian dollar down to lows not seen for over ten years.
Here are some plots of oil prices ($US per barrel, WTI) and how many US dollars one Canadian dollar buys. The monthly data are sourced from FRED. Notice the close correlation between the two data sets over the last half of the sample period.
As a starting point, I estimate a linear relationship between the two series.
The plot shows a positive correlation between oil prices and the exchange rate, but there seems to be some nonlinear behaviour towards the beginning and ending period of the data set.
Next, I estimate a quadratic fit. This doesn't look too different from the linear fit.
Perhaps a cubic fit is more appropriate. The cubic fit looks much better! Notice how the cubic curve fits the curvature at the beginning and ending period of the data.
Here are the model fits. The cubic curve fits the best.
|
linear |
quadratic |
cubic |
Adjusted R squared |
0.6511 |
0.6507 |
0.6923 |
Sigma |
0.0660 |
0.0660 |
0.0620 |
Here are some forecasts for each of the three regression models.
oil price |
linear |
quadratic |
cubic |
30.00 |
0.77 |
0.77 |
0.75 |
35.00 |
0.79 |
0.78 |
0.75 |
40.00 |
0.80 |
0.80 |
0.77 |
45.00 |
0.82 |
0.81 |
0.78 |
50.00 |
0.83 |
0.83 |
0.80 |
55.00 |
0.85 |
0.84 |
0.82 |
60.00 |
0.86 |
0.86 |
0.85 |
65.00 |
0.87 |
0.87 |
0.87 |
Today, oil is currently trading around $37 per barrel. An oil price of $35 per barrel produces an exchange rate forecast of 75 cents (using the cubic model). The actual exchange rate at time of writing is 74 cents. Overall, a fairly close fit.
The R script and data set are posted below.
#########################################################
# Economic forecasting and analysis
# Perry Sadorsky
# December 2015
# Oil prices and the Canadian dollar
##########################################################
rm(list=ls())
# load libraries
library(fpp)
as2_data <- read.csv("C:/econ 6210/6210f15/assignment 2/as2_data.csv")
View(as2_data)
df = as2_data
df = ts(df, start=1986, frequency=12)
plot(df[,-1], main="Oil prices and exchange rates", ylab = "", xlab = "")
oil = df[,"oil"]
fx = df[,"fx"]
# 5 year rolling correlations
rollout1 = rollapply(df[,-1], 60 ,function(x) cor(x[,1],x[,2]), by.column=FALSE,align="right")
rollout1 = na.omit(rollout1)
plot(rollout1,main="Rolling 5 year correlations between FX and Oil prices")
## linear fit
lm1 = lm(fx ~ oil)
summary(lm1)
str(summary(lm1))
rr = matrix(0,nrow=2, ncol=3)
rr[1,1] = summary(lm1)$adj.r.squared
rr[2,1] = summary(lm1)$sigma
par(mfrow=c(2,2))
plot(lm1)
par(mfrow=c(1,1))
plot(fx ~ oil, main="Linear fit",
ylab="$US/$C", xlab="Oil prices ($/bbl)")
abline(lm1)
oil_f = c(30, 35, 40, 45, 50, 55, 60, 65)
fcast1 <- forecast(lm1, newdata=data.frame(oil=oil_f))
fcast1
plot(fcast1, ylab="$US/$C", xlab="Oil prices ($/bbl)")
## quadratic fit
lm2 = lm(fx ~ oil + I(oil^2))
summary(lm2)
rr[1,2] = summary(lm2)$adj.r.squared
rr[2,2] = summary(lm2)$sigma
par(mfrow=c(2,2))
plot(lm2)
par(mfrow=c(1,1))
plot(fx ~ oil, main="Quadratic fit",ylab="$US/$C", xlab="Oil prices ($/bbl)")
curve( coef(lm2)[1] + coef(lm2)[2]*x + coef(lm2)[3]*x^2 , add=T )
fcast2 <- forecast(lm2, newdata=data.frame(oil=oil_f))
fcast2
## cubic fit
lm3 = lm(fx ~ oil +I(oil^2) +I(oil^3) )
summary(lm3)
rr[1,3] = summary(lm3)$adj.r.squared
rr[2,3] = summary(lm3)$sigma
par(mfrow=c(2,2))
plot(lm3)
par(mfrow=c(1,1))
plot(fx ~ oil, main="Cubic fit",ylab="$US/$C", xlab="Oil prices ($/bbl)")
curve( coef(lm3)[1] + coef(lm3)[2]*x + coef(lm3)[3]*x^2 + coef(lm3)[4]*x^3, add=T , col = "blue" )
fcast3 <- forecast(lm3, newdata=data.frame(oil=oil_f))
fcast3
fcast3$mean
tablef = cbind(oil_f, fcast1$mean, fcast2$mean, fcast3$mean)
colnames(tablef) = c("oil price", "linear", "quadratic", "cubic")
tablef
colnames(rr) = c("linear", "quadratic","cubic")
rownames(rr) = c("Adjusted R squared", "Sigma")
rr
date oil fx
1 1/1/1986 22.93 0.7107321
2 2/1/1986 15.46 0.7120986
3 3/1/1986 12.61 0.7138268
4 4/1/1986 12.84 0.7205130
5 5/1/1986 15.38 0.7269027
6 6/1/1986 13.43 0.7194762
7 7/1/1986 11.59 0.7242178
8 8/1/1986 15.10 0.7202017
9 9/1/1986 14.87 0.7208246
10 10/1/1986 14.90 0.7202017
11 11/1/1986 15.22 0.7213446
12 12/1/1986 16.11 0.7245852
13 1/1/1987 18.65 0.7349699
14 2/1/1987 17.75 0.7496252
15 3/1/1987 18.30 0.7579203
16 4/1/1987 18.68 0.7580352
17 5/1/1987 19.44 0.7456566
18 6/1/1987 20.07 0.7469934
19 7/1/1987 21.34 0.7540341
20 8/1/1987 20.31 0.7543754
21 9/1/1987 19.53 0.7602250
22 10/1/1987 19.86 0.7635336
23 11/1/1987 18.85 0.7594744
24 12/1/1987 17.28 0.7648184
25 1/1/1988 17.13 0.7779074
26 2/1/1988 16.80 0.7885192
27 3/1/1988 16.20 0.8005123
28 4/1/1988 17.86 0.8095200
29 5/1/1988 17.42 0.8082114
30 6/1/1988 16.53 0.8212878
31 7/1/1988 15.50 0.8281574
32 8/1/1988 15.52 0.8171938
33 9/1/1988 14.54 0.8151952
34 10/1/1988 13.77 0.8295313
35 11/1/1988 14.14 0.8206138
36 12/1/1988 16.38 0.8359806
37 1/1/1989 18.02 0.8394191
38 2/1/1989 17.94 0.8409722
39 3/1/1989 19.48 0.8365401
40 4/1/1989 21.07 0.8411844
41 5/1/1989 20.12 0.8385744
42 6/1/1989 20.05 0.8343067
43 7/1/1989 19.78 0.8409722
44 8/1/1989 18.58 0.8504848
45 9/1/1989 19.59 0.8454515
46 10/1/1989 20.10 0.8511363
47 11/1/1989 19.86 0.8549201
48 12/1/1989 21.10 0.8611039
49 1/1/1990 22.86 0.8532423
50 2/1/1990 22.11 0.8357710
51 3/1/1990 20.39 0.8474576
52 4/1/1990 18.43 0.8590327
53 5/1/1990 18.20 0.8512812
54 6/1/1990 16.70 0.8525149
55 7/1/1990 18.45 0.8643042
56 8/1/1990 27.31 0.8735150
57 9/1/1990 33.51 0.8633342
58 10/1/1990 36.04 0.8620690
59 11/1/1990 32.33 0.8594757
60 12/1/1990 27.28 0.8618461
61 1/1/1991 25.23 0.8650519
62 2/1/1991 20.48 0.8658758
63 3/1/1991 19.90 0.8641549
64 4/1/1991 20.83 0.8669267
65 5/1/1991 21.23 0.8696408
66 6/1/1991 20.19 0.8742023
67 7/1/1991 21.40 0.8700948
68 8/1/1991 21.69 0.8732099
69 9/1/1991 21.89 0.8795075
70 10/1/1991 23.23 0.8866034
71 11/1/1991 22.46 0.8847992
72 12/1/1991 19.50 0.8720677
73 1/1/1992 18.79 0.8642295
74 2/1/1992 19.01 0.8456660
75 3/1/1992 18.92 0.8383635
76 4/1/1992 20.23 0.8421762
77 5/1/1992 20.98 0.8339588
78 6/1/1992 22.39 0.8361204
79 7/1/1992 21.78 0.8386448
80 8/1/1992 21.34 0.8398421
81 9/1/1992 21.88 0.8179959
82 10/1/1992 21.69 0.8030194
83 11/1/1992 20.34 0.7890169
84 12/1/1992 19.41 0.7858546
85 1/1/1993 19.03 0.7825338
86 2/1/1993 20.09 0.7935248
87 3/1/1993 20.32 0.8018603
88 4/1/1993 20.25 0.7923302
89 5/1/1993 19.95 0.7875256
90 6/1/1993 19.09 0.7819220
91 7/1/1993 17.89 0.7800312
92 8/1/1993 18.01 0.7645260
93 9/1/1993 17.50 0.7567159
94 10/1/1993 18.15 0.7539772
95 11/1/1993 16.61 0.7590709
96 12/1/1993 14.52 0.7514277
97 1/1/1994 15.03 0.7591285
98 2/1/1994 14.78 0.7449344
99 3/1/1994 14.68 0.7329229
100 4/1/1994 16.42 0.7230658
101 5/1/1994 17.89 0.7242178
102 6/1/1994 19.06 0.7227522
103 7/1/1994 19.66 0.7232750
104 8/1/1994 18.38 0.7255315
105 9/1/1994 17.45 0.7385524
106 10/1/1994 17.72 0.7405762
107 11/1/1994 18.07 0.7327618
108 12/1/1994 17.16 0.7197869
109 1/1/1995 18.04 0.7076139
110 2/1/1995 18.57 0.7140307
111 3/1/1995 18.54 0.7103786
112 4/1/1995 19.90 0.7266386
113 5/1/1995 19.74 0.7348078
114 6/1/1995 18.45 0.7259528
115 7/1/1995 17.33 0.7346459
116 8/1/1995 18.02 0.7378985
117 9/1/1995 18.23 0.7402472
118 10/1/1995 17.43 0.7430525
119 11/1/1995 17.99 0.7388799
120 12/1/1995 19.03 0.7303002
121 1/1/1996 18.86 0.7315824
122 2/1/1996 19.09 0.7271670
123 3/1/1996 21.33 0.7322789
124 4/1/1996 23.50 0.7357269
125 5/1/1996 21.17 0.7303002
126 6/1/1996 20.42 0.7321716
127 7/1/1996 21.30 0.7300869
128 8/1/1996 21.90 0.7287567
129 9/1/1996 23.97 0.7302468
130 10/1/1996 24.88 0.7403020
131 11/1/1996 23.71 0.7473283
132 12/1/1996 25.23 0.7341066
133 1/1/1997 25.13 0.7410701
134 2/1/1997 22.18 0.7376807
135 3/1/1997 20.97 0.7285974
136 4/1/1997 19.70 0.7172572
137 5/1/1997 20.82 0.7244277
138 6/1/1997 19.26 0.7223868
139 7/1/1997 19.66 0.7259528
140 8/1/1997 19.95 0.7191658
141 9/1/1997 19.80 0.7208766
142 10/1/1997 21.33 0.7210325
143 11/1/1997 20.19 0.7078143
144 12/1/1997 18.33 0.7007217
145 1/1/1998 16.72 0.6940107
146 2/1/1998 16.06 0.6976420
147 3/1/1998 15.12 0.7059156
148 4/1/1998 15.35 0.6993985
149 5/1/1998 14.91 0.6919458
150 6/1/1998 13.72 0.6823610
151 7/1/1998 14.17 0.6725402
152 8/1/1998 13.47 0.6516356
153 9/1/1998 15.03 0.6571166
154 10/1/1998 14.46 0.6471654
155 11/1/1998 13.00 0.6491820
156 12/1/1998 11.35 0.6479622
157 1/1/1999 12.52 0.6581545
158 2/1/1999 12.01 0.6676905
159 3/1/1999 14.68 0.6589352
160 4/1/1999 17.31 0.6719979
161 5/1/1999 17.72 0.6844159
162 6/1/1999 17.92 0.6805036
163 7/1/1999 20.10 0.6715917
164 8/1/1999 21.28 0.6697027
165 9/1/1999 23.80 0.6770022
166 10/1/1999 22.69 0.6767731
167 11/1/1999 25.00 0.6814774
168 12/1/1999 26.10 0.6792555
169 1/1/2000 27.26 0.6903217
170 2/1/2000 29.37 0.6890849
171 3/1/2000 29.84 0.6845564
172 4/1/2000 25.72 0.6807815
173 5/1/2000 28.79 0.6685833
174 6/1/2000 31.82 0.6770481
175 7/1/2000 29.70 0.6766816
176 8/1/2000 31.26 0.6743998
177 9/1/2000 33.88 0.6727664
178 10/1/2000 33.11 0.6611570
179 11/1/2000 34.42 0.6482562
180 12/1/2000 28.44 0.6570734
181 1/1/2001 29.59 0.6652475
182 2/1/2001 29.61 0.6572029
183 3/1/2001 27.25 0.6415603
184 4/1/2001 27.49 0.6419309
185 5/1/2001 28.63 0.6488872
186 6/1/2001 27.60 0.6559528
187 7/1/2001 26.43 0.6532532
188 8/1/2001 27.37 0.6493928
189 9/1/2001 26.20 0.6377958
190 10/1/2001 22.17 0.6362537
191 11/1/2001 19.64 0.6280618
192 12/1/2001 19.39 0.6333924
193 1/1/2002 19.72 0.6251172
194 2/1/2002 20.72 0.6264094
195 3/1/2002 24.53 0.6298419
196 4/1/2002 26.18 0.6323111
197 5/1/2002 27.04 0.6450781
198 6/1/2002 25.52 0.6528267
199 7/1/2002 26.97 0.6469979
200 8/1/2002 28.39 0.6371862
201 9/1/2002 29.66 0.6344775
202 10/1/2002 28.84 0.6337136
203 11/1/2002 26.35 0.6363347
204 12/1/2002 29.46 0.6413545
205 1/1/2003 32.95 0.6487609
206 2/1/2003 35.83 0.6613319
207 3/1/2003 33.51 0.6774609
208 4/1/2003 28.17 0.6857770
209 5/1/2003 28.11 0.7225434
210 6/1/2003 30.66 0.7393715
211 7/1/2003 30.76 0.7235366
212 8/1/2003 31.57 0.7161785
213 9/1/2003 28.31 0.7334605
214 10/1/2003 30.34 0.7563724
215 11/1/2003 31.11 0.7616146
216 12/1/2003 32.13 0.7617307
217 1/1/2004 34.31 0.7717240
218 2/1/2004 34.69 0.7519362
219 3/1/2004 36.74 0.7526720
220 4/1/2004 36.75 0.7451565
221 5/1/2004 40.28 0.7252158
222 6/1/2004 38.03 0.7364855
223 7/1/2004 40.78 0.7561437
224 8/1/2004 44.90 0.7617887
225 9/1/2004 45.94 0.7763372
226 10/1/2004 53.28 0.8019889
227 11/1/2004 48.47 0.8355615
228 12/1/2004 43.15 0.8204118
229 1/1/2005 46.84 0.8164598
230 2/1/2005 48.15 0.8063866
231 3/1/2005 54.19 0.8223684
232 4/1/2005 52.98 0.8091270
233 5/1/2005 49.83 0.7964954
234 6/1/2005 56.35 0.8063216
235 7/1/2005 59.00 0.8177284
236 8/1/2005 64.99 0.8303579
237 9/1/2005 65.59 0.8491127
238 10/1/2005 62.26 0.8493290
239 11/1/2005 58.32 0.8463817
240 12/1/2005 59.41 0.8609557
241 1/1/2006 65.49 0.8641549
242 2/1/2006 61.63 0.8703978
243 3/1/2006 62.69 0.8640802
244 4/1/2006 69.44 0.8740495
245 5/1/2006 70.84 0.9009009
246 6/1/2006 70.95 0.8979079
247 7/1/2006 74.41 0.8854259
248 8/1/2006 73.04 0.8942944
249 9/1/2006 63.80 0.8959771
250 10/1/2006 58.89 0.8861320
251 11/1/2006 59.08 0.8803592
252 12/1/2006 61.96 0.8671523
253 1/1/2007 54.51 0.8501233
254 2/1/2007 59.28 0.8539710
255 3/1/2007 60.44 0.8560178
256 4/1/2007 63.98 0.8810573
257 5/1/2007 63.46 0.9131586
258 6/1/2007 67.49 0.9388790
259 7/1/2007 74.12 0.9521996
260 8/1/2007 72.36 0.9452689
261 9/1/2007 79.92 0.9739944
262 10/1/2007 85.80 1.0252204
263 11/1/2007 94.77 1.0339123
264 12/1/2007 91.69 0.9979044
265 1/1/2008 92.97 0.9901970
266 2/1/2008 95.39 1.0014020
267 3/1/2008 105.45 0.9971084
268 4/1/2008 112.58 0.9864852
269 5/1/2008 125.40 1.0007005
270 6/1/2008 133.88 0.9836711
271 7/1/2008 133.37 0.9871668
272 8/1/2008 116.67 0.9492169
273 9/1/2008 104.11 0.9450009
274 10/1/2008 76.61 0.8440956
275 11/1/2008 57.31 0.8216252
276 12/1/2008 41.12 0.8105698
277 1/1/2009 41.71 0.8164598
278 2/1/2009 39.09 0.8030838
279 3/1/2009 47.94 0.7908264
280 4/1/2009 49.65 0.8168600
281 5/1/2009 59.03 0.8674532
282 6/1/2009 69.64 0.8877841
283 7/1/2009 64.15 0.8905513
284 8/1/2009 71.05 0.9197940
285 9/1/2009 69.41 0.9245562
286 10/1/2009 75.72 0.9481369
287 11/1/2009 77.99 0.9440196
288 12/1/2009 74.47 0.9490367
289 1/1/2010 78.33 0.9580379
290 2/1/2010 76.39 0.9458948
291 3/1/2010 81.20 0.9776127
292 4/1/2010 84.29 0.9948269
293 5/1/2010 73.74 0.9612612
294 6/1/2010 75.34 0.9637625
295 7/1/2010 76.32 0.9595087
296 8/1/2010 76.60 0.9611688
297 9/1/2010 75.24 0.9680542
298 10/1/2010 81.89 0.9824148
299 11/1/2010 84.25 0.9872643
300 12/1/2010 89.15 0.9919651
301 1/1/2011 89.17 1.0061374
302 2/1/2011 88.58 1.0125557
303 3/1/2011 102.86 1.0239607
304 4/1/2011 109.53 1.0438413
305 5/1/2011 100.90 1.0330579
306 6/1/2011 96.26 1.0239607
307 7/1/2011 97.30 1.0467916
308 8/1/2011 86.33 1.0186411
309 9/1/2011 85.52 0.9975062
310 10/1/2011 86.32 0.9805844
311 11/1/2011 97.16 0.9758002
312 12/1/2011 98.56 0.9770396
313 1/1/2012 100.27 0.9871668
314 2/1/2012 102.20 1.0033109
315 3/1/2012 106.16 1.0062387
316 4/1/2012 103.32 1.0072522
317 5/1/2012 94.66 0.9903932
318 6/1/2012 82.30 0.9727626
319 7/1/2012 87.90 0.9859988
320 8/1/2012 94.13 1.0076582
321 9/1/2012 94.51 1.0221813
322 10/1/2012 89.49 1.0129660
323 11/1/2012 86.53 1.0030090
324 12/1/2012 87.86 1.0103051
325 1/1/2013 94.76 1.0079629
326 2/1/2013 95.31 0.9902951
327 3/1/2013 92.94 0.9761812
328 4/1/2013 92.02 0.9816433
329 5/1/2013 94.51 0.9807768
330 6/1/2013 95.77 0.9695559
331 7/1/2013 104.67 0.9613536
332 8/1/2013 106.57 0.9608917
333 9/1/2013 106.29 0.9669310
334 10/1/2013 100.54 0.9649715
335 11/1/2013 93.86 0.9536525
336 12/1/2013 97.63 0.9399380
337 1/1/2014 94.62 0.9140768
338 2/1/2014 100.82 0.9046499
339 3/1/2014 100.80 0.9003331
340 4/1/2014 102.07 0.9097525
341 5/1/2014 102.18 0.9179365
342 6/1/2014 105.79 0.9233610
343 7/1/2014 103.59 0.9311854
344 8/1/2014 96.54 0.9152480
345 9/1/2014 93.21 0.9081827
346 10/1/2014 84.40 0.8919015
347 11/1/2014 75.79 0.8830022
348 12/1/2014 59.29 0.8671523
349 1/1/2015 47.22 0.8249464
350 2/1/2015 50.58 0.8000640
351 3/1/2015 47.82 0.7925186
352 4/1/2015 54.45 0.8105698
353 5/1/2015 59.27 0.8212878
354 6/1/2015 59.82 0.8087343
355 7/1/2015 50.90 0.7774236
356 8/1/2015 42.87 0.7606298
357 9/1/2015 45.48 0.7538067
358 10/1/2015 46.22 0.7649939
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