Help me interpret my results for my second report should consist of an application of the time series econometrics learnt in Modules 4-5 to your data. Full submission should consist of a document containing the introduction, literature review, methods and data and preliminary results; the file containing your data and your computer codes. Please present the results (tables, plots, etc. in your report).

USL is United States 12 Month Oil Fund LP.
FUT is Future oil Prices.

> # Load datasets
> Fut <- read_excel("OILNF_US0_NEW.xlsx", sheet = "Fut", col_types = c("date", "numeric"))
> USL <- read_excel("OILNF_US0_NEW.xlsx", sheet = "USL", col_types = c("date", "numeric"))
Warning messages:
1: Expecting numeric in B3150 / R3150C2: got '#N/A N/A' 
2: Expecting numeric in B3151 / R3151C2: got '#N/A N/A' 
3: Expecting numeric in B3152 / R3152C2: got '#N/A N/A' 
> # Transform data to logarithmic scale and add time column
> log_data <- Fut %>%
+   arrange(Dates) %>% 
+   full_join(USL, by = "Dates") %>%
+   na.omit() %>%
+   mutate(
+     log_Fut = log(Fut),
+     log_USL = log(USL),
+     Time = row_number()  # Add an incremental time column
+   ) %>%
+   select(Time, Dates, log_Fut, log_USL)
> # Compute first differences of log-transformed data
> diff_log_data <- log_data %>%
+   arrange(Dates) %>%
+   mutate(
+     diff_log_Fut = c(NA, diff(log_Fut)),
+     diff_log_USL = c(NA, diff(log_USL))
+   ) %>%
+   drop_na() %>%
+   select(Time, Dates, diff_log_Fut, diff_log_USL)
> # Conduct Dickey-Fuller tests on log-transformed data
> dicky_fuller.trend <- lapply(log_data[, c("log_Fut", "log_USL")], 
+                              function(x) ur.df(as.vector(x), type = "trend", selectlags = "BIC"))
> dicky_fuller.constant <- lapply(log_data[, c("log_Fut", "log_USL")], 
+                                 function(x) ur.df(as.vector(x), type = "drift", selectlags = "BIC"))
> dicky_fuller.none <- lapply(log_data[, c("log_Fut", "log_USL")], 
+                             function(x) ur.df(as.vector(x), type = "none", selectlags = "BIC"))
> 
> # Print summaries of Dickey-Fuller tests
> lapply(dicky_fuller.trend, summary)
$log_Fut

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression trend 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.132672 -0.011079  0.000854  0.011452  0.157751 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  1.454e-02  6.804e-03   2.137   0.0327 * 
z.lag.1     -3.245e-03  1.484e-03  -2.186   0.0289 * 
tt          -5.481e-07  5.420e-07  -1.011   0.3120   
z.diff.lag  -5.053e-02  1.782e-02  -2.836   0.0046 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02362 on 3142 degrees of freedom
Multiple R-squared:  0.004243,	Adjusted R-squared:  0.003292 
F-statistic: 4.463 on 3 and 3142 DF,  p-value: 0.003915


Value of test-statistic is: -2.1862 1.627 2.3997 

Critical values for test statistics: 
      1pct  5pct 10pct
tau3 -3.96 -3.41 -3.12
phi2  6.09  4.68  4.03
phi3  8.27  6.25  5.34


$log_USL

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression trend 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.106327 -0.008434  0.000813  0.009516  0.084499 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)  
(Intercept)  9.926e-03  5.406e-03   1.836   0.0664 .
z.lag.1     -2.593e-03  1.344e-03  -1.929   0.0538 .
tt          -7.918e-07  5.879e-07  -1.347   0.1782  
z.diff.lag  -3.500e-02  1.783e-02  -1.963   0.0497 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01826 on 3142 degrees of freedom
Multiple R-squared:  0.00253,	Adjusted R-squared:  0.001578 
F-statistic: 2.657 on 3 and 3142 DF,  p-value: 0.04682


Value of test-statistic is: -1.9289 1.4759 1.9049 

Critical values for test statistics: 
      1pct  5pct 10pct
tau3 -3.96 -3.41 -3.12
phi2  6.09  4.68  4.03
phi3  8.27  6.25  5.34


> lapply(dicky_fuller.constant, summary)
$log_Fut

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression drift 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.131682 -0.011082  0.000761  0.011450  0.159001 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)   
(Intercept)  0.010373   0.005416   1.915  0.05554 . 
z.lag.1     -0.002468   0.001270  -1.943  0.05206 . 
z.diff.lag  -0.050965   0.017811  -2.862  0.00424 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02362 on 3143 degrees of freedom
Multiple R-squared:  0.003919,	Adjusted R-squared:  0.003285 
F-statistic: 6.183 on 2 and 3143 DF,  p-value: 0.002089


Value of test-statistic is: -1.9434 1.9292 

Critical values for test statistics: 
      1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78


$log_USL

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression drift 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.106083 -0.008531  0.000711  0.009510  0.084921 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)  
(Intercept)  0.0037347  0.0028437   1.313   0.1892  
z.lag.1     -0.0011581  0.0008198  -1.413   0.1579  
z.diff.lag  -0.0358395  0.0178196  -2.011   0.0444 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01826 on 3143 degrees of freedom
Multiple R-squared:  0.001954,	Adjusted R-squared:  0.001319 
F-statistic: 3.077 on 2 and 3143 DF,  p-value: 0.04622


Value of test-statistic is: -1.4127 1.3068 

Critical values for test statistics: 
      1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78


> lapply(dicky_fuller.none, summary)
$log_Fut

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression none 


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.130733 -0.011038  0.000468  0.011342  0.160750 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)   
z.lag.1    -4.305e-05  9.882e-05  -0.436  0.66312   
z.diff.lag -5.216e-02  1.781e-02  -2.929  0.00342 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02363 on 3144 degrees of freedom
Multiple R-squared:  0.00278,	Adjusted R-squared:  0.002145 
F-statistic: 4.382 on 2 and 3144 DF,  p-value: 0.01258


Value of test-statistic is: -0.4356 

Critical values for test statistics: 
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62


$log_USL

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression none 


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.106740 -0.008536  0.000660  0.009450  0.085543 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)  
z.lag.1    -8.850e-05  9.389e-05  -0.943   0.3460  
z.diff.lag -3.633e-02  1.782e-02  -2.039   0.0415 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01827 on 3144 degrees of freedom
Multiple R-squared:  0.001589,	Adjusted R-squared:  0.0009537 
F-statistic: 2.502 on 2 and 3144 DF,  p-value: 0.08212


Value of test-statistic is: -0.9426 

Critical values for test statistics: 
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62


> 
> # Conduct Dickey-Fuller tests on first-difference data
> diff_df.trend <- lapply(diff_log_data[, c("diff_log_Fut", "diff_log_USL")], 
+                         function(x) ur.df(as.vector(x), type = "trend", selectlags = "BIC"))
> diff_df.constant <- lapply(diff_log_data[, c("diff_log_Fut", "diff_log_USL")], 
+                            function(x) ur.df(as.vector(x), type = "drift", selectlags = "BIC"))
> diff_df.none <- lapply(diff_log_data[, c("diff_log_Fut", "diff_log_USL")], 
+                        function(x) ur.df(as.vector(x), type = "none", selectlags = "BIC"))
> 
> # Print summaries of Dickey-Fuller tests on differences
> lapply(diff_df.trend, summary)
$diff_log_Fut

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression trend 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.128257 -0.011036  0.000608  0.011333  0.155890 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -2.274e-04  8.429e-04  -0.270  0.78737    
z.lag.1     -1.102e+00  2.585e-02 -42.618  < 2e-16 ***
tt           6.542e-08  4.639e-07   0.141  0.88786    
z.diff.lag   4.711e-02  1.782e-02   2.644  0.00824 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02362 on 3141 degrees of freedom
Multiple R-squared:  0.5272,	Adjusted R-squared:  0.5267 
F-statistic:  1167 on 3 and 3141 DF,  p-value: < 2.2e-16


Value of test-statistic is: -42.6185 605.4453 908.1679 

Critical values for test statistics: 
      1pct  5pct 10pct
tau3 -3.96 -3.41 -3.12
phi2  6.09  4.68  4.03
phi3  8.27  6.25  5.34


$diff_log_USL

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression trend 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.106586 -0.008681  0.000631  0.009393  0.085398 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -4.206e-04  6.523e-04  -0.645    0.519    
z.lag.1     -1.030e+00  2.569e-02 -40.106   <2e-16 ***
tt           1.058e-07  3.589e-07   0.295    0.768    
z.diff.lag  -6.015e-03  1.784e-02  -0.337    0.736    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01827 on 3141 degrees of freedom
Multiple R-squared:  0.5182,	Adjusted R-squared:  0.5177 
F-statistic:  1126 on 3 and 3141 DF,  p-value: < 2.2e-16


Value of test-statistic is: -40.1064 536.1738 804.2607 

Critical values for test statistics: 
      1pct  5pct 10pct
tau3 -3.96 -3.41 -3.12
phi2  6.09  4.68  4.03
phi3  8.27  6.25  5.34


> lapply(diff_df.constant, summary)
$diff_log_Fut

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression drift 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.128341 -0.011063  0.000597  0.011342  0.155806 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.0001244  0.0004211  -0.295  0.76766    
z.lag.1     -1.1018014  0.0258488 -42.625  < 2e-16 ***
z.diff.lag   0.0471053  0.0178177   2.644  0.00824 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02361 on 3142 degrees of freedom
Multiple R-squared:  0.5272,	Adjusted R-squared:  0.5269 
F-statistic:  1752 on 2 and 3142 DF,  p-value: < 2.2e-16


Value of test-statistic is: -42.6249 908.4414 

Critical values for test statistics: 
      1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78


$diff_log_USL

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression drift 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.106745 -0.008659  0.000601  0.009359  0.085443 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.0002541  0.0003259  -0.780    0.436    
z.lag.1     -1.0300903  0.0256809 -40.111   <2e-16 ***
z.diff.lag  -0.0060460  0.0178361  -0.339    0.735    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01827 on 3142 degrees of freedom
Multiple R-squared:  0.5182,	Adjusted R-squared:  0.5179 
F-statistic:  1690 on 2 and 3142 DF,  p-value: < 2.2e-16


Value of test-statistic is: -40.1111 804.4511 

Critical values for test statistics: 
      1pct  5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1  6.43  4.59  3.78


> lapply(diff_df.none, summary)
$diff_log_Fut

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression none 


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.128467 -0.011187  0.000472  0.011218  0.155686 

Coefficients:
           Estimate Std. Error t value Pr(>|t|)    
z.lag.1    -1.10175    0.02584 -42.630  < 2e-16 ***
z.diff.lag  0.04708    0.01781   2.643  0.00827 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02361 on 3143 degrees of freedom
Multiple R-squared:  0.5272,	Adjusted R-squared:  0.5269 
F-statistic:  1752 on 2 and 3143 DF,  p-value: < 2.2e-16


Value of test-statistic is: -42.6301 

Critical values for test statistics: 
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62


$diff_log_USL

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression none 


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.106998 -0.008917  0.000341  0.009105  0.085184 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
z.lag.1    -1.029696   0.025674  -40.11   <2e-16 ***
z.diff.lag -0.006246   0.017833   -0.35    0.726    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01827 on 3143 degrees of freedom
Multiple R-squared:  0.5181,	Adjusted R-squared:  0.5178 
F-statistic:  1690 on 2 and 3143 DF,  p-value: < 2.2e-16


Value of test-statistic is: -40.106 

Critical values for test statistics: 
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62