R语言 ur.df函数

It seems the creators of this particular R command presume one is familiar with the original Dickey-Fuller formulae, so did not provide the relevant documentation for how to interpret the values. I found that Enders was an incredibly helpful resource (Applied Econometric Time Series 3e, 2010, p. 206-209--I imagine other editions would also be fine). Below I'll use data from the URCA package, real income in Denmark as an example.

> income <- ts(denmark$LRY)

It might be useful to first describe the 3 different formulae Dickey-Fuller used to get different hypotheses, since these match the ur.df "type" options. Enders specifies that in all of these 3 cases, the consistent term used is gamma, the coefficient for the previous value of y, the lag term. If gamma=0, then there is a unit root (random walk, nonstationary). Where the null hypothesis is gamma=0, if p<0.05, then we reject the null (at the 95% level), and presume there is no unit root. If we fail to reject the null (p>0.05) then we presume a unit root exists. From here, we can proceed to interpreting the tau's and phi's.

1) type="none": Δ𝑦(𝑡)=𝛾∗𝑦(𝑡−1)+𝑒(𝑡)Δy(t)=γ∗y(t−1)+e(t) (formula from Enders p. 208)

(where 𝑒(𝑡)e(t) is the error term, presumed to be white noise; 𝛾=𝑎−1γ=a−1 from 𝑦=𝑎∗𝑦(𝑡−1)+𝑒(𝑡)y=a∗y(t−1)+e(t); 𝑦(𝑡−1)y(t−1) refers to the previous value of y, so is the lag term)

For type= "none," tau (or tau1 in R output) is the null hypothesis for gamma = 0. Using the Denmark income example, I get "Value of test-statistic is 0.7944" and the "Critical values for test statistics are: tau1 -2.6 -1.95 -1.61. Given that the test statistic is within the all 3 regions (1%, 5%, 10%) where we fail to reject the null, we should presume the data is a random walk, ie that a unit root is present. In this case, the tau1 refers to the gamma = 0 hypothesis. The "z.lag1" is the gamma term, the coefficient for the lag term (y(t-1)), which is p=0.431, which we fail to reject as significant, simply implying that gamma isn't statistically significant to this model. Here is the output from R

> summary(ur.df(y=income, type = "none",lags=1))
> 
> ############################################### 
> # 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 
> -0.044067 -0.016747 -0.006596  0.010305  0.085688 
> 
> Coefficients:
>             Estimate Std. Error t value Pr(>|t|)
> z.lag.1    0.0004636  0.0005836   0.794    0.431
> z.diff.lag 0.1724315  0.1362615   1.265    0.211
> 
> Residual standard error: 0.0251 on 51 degrees of freedom
> Multiple R-squared:  0.04696,   Adjusted R-squared:  0.009589 
> F-statistic: 1.257 on 2 and 51 DF,  p-value: 0.2933
> 
> 
> Value of test-statistic is: 0.7944 
> 
> Critical values for test statistics: 
>      1pct  5pct 10pct
> tau1 -2.6 -1.95 -1.61

2) type = "drift" (your specific question above): : Δ𝑦(𝑡)=𝑎0+𝛾∗𝑦(𝑡−1)+𝑒(𝑡)Δy(t)=a0+γ∗y(t−1)+e(t) (formula from Enders p. 208)

(where a0 is "a sub-zero" and refers to the constant, or drift term) Here is where the output interpretation gets trickier. "tau2" is still the 𝛾=0γ=0 null hypothesis. In this case, where the first test statistic = -1.4462 is within the region of failing to reject the null, we should again presume a unit root, that 𝛾=0γ=0.
The phi1 term refers to the second hypothesis, which is a combined null hypothesis of a0 = gamma = 0. This means that BOTH of the values are tested to be 0 at the same time. If p<0.05, we reject the null, and presume that AT LEAST one of these is false--i.e. one or both of the terms a0 or gamma are not 0. Failing to reject this null implies that BOTH a0 AND gamma = 0, implying 1) that gamma=0 therefore a unit root is present, AND 2) a0=0, so there is no drift term. Here is the R output

> summary(ur.df(y=income, type = "drift",lags=1))
> 
> ############################################### 
> # 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 
> -0.041910 -0.016484 -0.006994  0.013651  0.074920 
> 
> Coefficients:
>             Estimate Std. Error t value Pr(>|t|)
> (Intercept)  0.43453    0.28995   1.499    0.140
> z.lag.1     -0.07256    0.04873  -1.489    0.143
> z.diff.lag   0.22028    0.13836   1.592    0.118
> 
> Residual standard error: 0.0248 on 50 degrees of freedom
> Multiple R-squared:  0.07166,   Adjusted R-squared:  0.03452 
> F-statistic:  1.93 on 2 and 50 DF,  p-value: 0.1559
> 
> 
> Value of test-statistic is: -1.4891 1.4462 
> 
> Critical values for test statistics: 
>       1pct  5pct 10pct
> tau2 -3.51 -2.89 -2.58
> phi1  6.70  4.71  3.86

3) Finally, for the type="trend": Δ𝑦(𝑡)=𝑎0+𝑔𝑎𝑚𝑚𝑎∗𝑦(𝑡−1)+𝑎2(𝑡)+𝑒(𝑡)Δy(t)=a0+gamma∗y(t−1)+a2(t)+e(t) (formula from Enders p. 208)

(where a2(t) is a time trend term) The hypotheses (from Enders p. 208) are as follows: tau: gamma=0 phi3: gamma = a2 = 0 phi2: a0 = gamma = a2 = 0 This is similar to the R output. In this case, the test statistics are -2.4216 2.1927 2.9343 In all of these cases, these fall within the "fail to reject the null" zones (see critical values below). What tau3 implies, as above, is that we fail to reject the null of unit root, implying a unit root is present. Failing to reject phi3 implies two things: 1) gamma = 0 (unit root) AND 2) there is no time trend term, i.e., a2=0. If we rejected this null, it would imply that one or both of these terms was not 0. Failing to reject phi2 implies 3 things: 1) gamma = 0 AND 2) no time trend term AND 3) no drift term, i.e. that gamma =0, that a0 = 0, and that a2 = 0. Rejecting this null implies that one, two, OR all three of these terms was NOT zero.
Here is the R output

> summary(ur.df(y=income, type = "trend",lags=1))
> 
> ############################################### 
> # 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 
> -0.036693 -0.016457 -0.000435  0.014344  0.074299 
> 
> Coefficients:
>               Estimate Std. Error t value Pr(>|t|)  
> (Intercept)  1.0369478  0.4272693   2.427   0.0190 *
> z.lag.1     -0.1767666  0.0729961  -2.422   0.0192 *
> tt           0.0006299  0.0003348   1.881   0.0659 .
> z.diff.lag   0.2557788  0.1362896   1.877   0.0665 .
> ---
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> 
> Residual standard error: 0.02419 on 49 degrees of freedom
> Multiple R-squared:  0.1342,    Adjusted R-squared:  0.08117 
> F-statistic: 2.531 on 3 and 49 DF,  p-value: 0.06785
> 
> 
> Value of test-statistic is: -2.4216 2.1927 2.9343 
> 
> Critical values for test statistics: 
>       1pct  5pct 10pct
> tau3 -4.04 -3.45 -3.15
> phi2  6.50  4.88  4.16
> phi3  8.73  6.49  5.47

In your specific example above, for the d.Aus data, since both of the test statistics are inside of the "fail to reject" zone, it implies that gamma=0 AND a0 = 0, meaning that there is a unit root, but no drift term.