Duration data measure the time an individual spends in a specific state, denoted duration time, and interest lies in modeling the time until leaving that state; see, e.g., [1, 2]. A typical example is movement between labour market states of employment. When considering unemployment, there are factors not obvious how to incorporate; for example, unemployment spells often are considerably longer in times of recession than during a boom, and representing the state of the business cycle as dichotomous would not be appropriate. The economy is not always in the extremes but often somewhere in between. In the standard regression and time series literature, a class of models including this feature is the smooth transition model, with a smooth transition between the different states. In addition, interest in nonlinear models has, in recent years, been steadily increasing and nonlinear models seem to outperform linear ones in terms of fit and forecasting performance, see, e.g., [3, 4].

In this paper, we introduce the smooth transition duration model. A continuum of states is assumed, where an observed deterministic transition variable controls for the transition from one regime to another. We allow both time-varying covariates and the duration of time to vary with smooth transitions over different regimes. The smooth transition duration model is estimated using maximum likelihood, and the likelihood, scores, and information matrix are given. Proof of asymptotic normality of the parameter estimates is also presented. As the model is unidentified if there is no smooth transition, it is crucial to test for nonlinearity and an LM test, in the spirit of [5, 6], is developed for smooth transition durations. Finally, a Monte Carlo simulation is carried out to analyse the proposed test’s size and size-adjusted power.

We exemplify the use of our proposed model with an empirical application, where the effects of deregulation in the energy market in Queensland, Australia, are assessed. More specifically, we model the time between abnormal electricity spot prices. Since short-lived price increases are suspected to be a result of strategic bidding behaviour by electricity generators to force the spot price up, it is of interest to examine whether this behaviour is affected by regulatory changes. The results support the existence of a gradual change in the appearance of abnormal price increases. An explicit aim is to investigate if there has been a change in occurrences of abnormal spot price (i.e. shorter time span between events) in relation to the deregulation of the electricity market. This has important policy implications as increased occurrences would suggest that there is a need for the authorities to monitor the market more closely for the purpose of a more efficient electricity price market.

Hurn et al. [7] analysed the same data (but with a shorter time span) using a smooth transition logit model where they model the probability of an extreme price event. Analysing time between events yields additional valuable information and relaxed assumptions. For example, the conditional independence assumption that the logit model assumes is not needed for our model, see e.g. [8].

The rest of the paper is organized as follows. Section 2 introduces the smooth transition duration model, the estimation method and linearity tests, where the performance is examined with a Monte Carlo simulation. Section 3 presents the empirical application. Section 4 concludes.

Let T be an absolutely continuous, non-negative random variable representing time until an event occurs, also called the duration, with distribution function \(F(t) = P(T \le t)\) and density \(f(t) = dF(t)/dt\). The distribution of T can then be uniquely characterized by the hazard function, specified as

The hazard function represents the instantaneous rate at which events occur if an event has not yet occurred at time t, and it fully specifies the distribution and determines the density and the survivor functions [1].

When modelling the duration experience, there are explanatory variables upon which the duration time might depend. Assuming the rate to be a function of derived covariates, the functional form of the hazard function can be specified as

where \(\textbf{x}= (\textbf{x}_{1},..., \textbf{x}_{p})\) is a vector of independent observable variables and \(\varvec{\beta }\) is a vector of model parameters. The function \(\lambda _0(t)\) is the baseline hazard. Different assumptions about the hazard can be made; in the simplest case, it is assumed to be constant. A generalization allows for a power dependence on duration time, such that

where \(\alpha \) parameterizes the duration dependence. The hazard rate decreases with duration time if \(\alpha < 1\) and increases if \(\alpha > 1\). This is the hazard function for the Weibull distribution. The Weibull distribution is very flexible and a standard choice in duration analysis, see e.g. [9].

It is of interest to allow the hazard rate to change due to, for example, an unknown threshold of a covariate or the timing of the event. Survival models including change-points according to a threshold in a covariate have been studied, see, among others, [10, 11] for semi-parametric approaches. Cases where the duration dependence changes at a threshold have also been evaluated, see, for example, [12, 13].

This paper contributes to the existing literature by allowing the duration to vary with smooth transitions over two or more regimes. The transitions occur at unknown change points, and depends on an observed transition variable. This results in a hazard rate model expressed as

where the logistic transition function \(G(s; \textbf{c}, \gamma )\) has the general form

with \(c=(c_1,..., c_K) \) being a vector of location parameters with \(c_1 \le \cdots \le c_K\) and \(\gamma > 0 \) controls the slope of the function. The transition function is a bounded function of the transition variable s. The transition variable is an exogenous stationary or deterministic variable and can, for example, be rescaled calendar time, such that \(s \in (0,1)\).

The cumulative hazard function of the smooth transition duration model is

leading to a Weibull form with the distribution function

The conditional density function of T given the covariates can then be specified as

An important generalization of the hazard model in Eq. (4) allows the covariates to be time-varying. We let each duration time be viewed as multiple records, where each record corresponds to an interval in which the value of the covariate is constant. This generalization imposes the need to handle both censored and truncated records.

Having a sample of \(j=1,...,n\) observations on durations, each with \(n_j\) records, and letting \(N=n_1+n_2+ \cdots +n_n\), leads to a total of N observations on records. Letting \(i=1,...,N\), then the vector of covariates \(\textbf{x}_i\) is assumed to be fixed for each observation i. Let \(c_i= 1\) if record i is an observed event and denote the observation censored, with \(c_i =0\), otherwise. The ith survival time, denoted by \(t_i\), is only observed conditional on surviving until time point \(r_i\), where \(r_i<t_i\). Also, let \(T_i\) denote the calendar time corresponding to i and let T denote the total number of calendar time points. As above, \(s_i\) is the transition variable at time point i.

Letting

for \(i=1,..., N\), the log likelihood of the smooth transition duration model is

where \(\varvec{\theta }= (\varvec{\beta }, \, \varvec{\psi }, \,\gamma ,\, c)\) is of dimension \(2(p + 1) + 2\) if \(\textbf{x}_i\) is p-dimensional [14]. When estimating the model, the slope parameter \(\gamma \) is replaced with \(e^{\eta }\). As described in [7], the gain is twofold. As the parameter to be estimated is now \(\eta \in (-\infty , \infty )\), the positive restriction on the slope parameter is avoided. At the same time, the uncertainty about the transition length is decreased in the case of a large value of the true \(\gamma \).

The score is

where

and

The information matrix can then be specified as

with

Assuming that the transition variable is time, the model becomes asymptotically unidentified, as mentioned in [7]. In such case as \(T \rightarrow \infty \), the proportion of observations in the first regime goes to zero. The parameter vector \(\varvec{\beta }\), governing the first regime, and the parameters \((\gamma ,c)\) vanish from the model and become unidentified. Using rescaled time \(s_i=t_i/T\) makes the model identified, but triangular array asymptotics (i.e., N and T both tend to infinity) need to be used instead of standard asymptotics, see [15]. Following [7, 16], consistency and asymptotic normality of the maximum likelihood estimators can be proven with the following assumptions:

A1: The true parameter vector is given by \(\varvec{\theta }_0 \in \Theta \) which is a compact space.

A2. The slope parameter \(\gamma _0\) satisfies \(\gamma _0 > 0\) and \(\varvec{\psi }_0 \ne \varvec{\beta }_0\) with \(\varvec{\psi }_0 \ne \varvec{0}\).

A3. The matrix \(M=\mathbb {E}\textbf{x}\textbf{x}^{\prime }\) is positive definite and \(\int e^{(\textbf{x})p(\textbf{x})}d\textbf{x}< \infty \) where \(p(\textbf{x})\) is the density of \(\textbf{x}\).

A4. \(\text {lim}_{T \rightarrow \infty } N/T = p \le p_0 < \infty \), i.e. that the ratio of events to the total of time points is asymptotically a finite constant and that \(T \rightarrow \infty \) implies \(N \rightarrow \infty \).

A5. The duration \( t_i/T \rightarrow 0\), \(i=1,...,N\) as \(T \rightarrow \infty \).

A6. The derivatives of the likelihood satisfies the martingale property \(\mathbb {E}\partial l_t(\varvec{\theta })/\partial \varvec{\theta }|\mathbb {F}_{t-1}=0\), where \(\mathbb {F}_{t}\) is the \(\sigma \)-algebra at t

See Appendix for proof.

The smooth transition duration model specified in Eq. (4) is linear if \(\varvec{\psi }= \varvec{0}\). In that case, the parameter \(\gamma \) is not identified, and the parameters cannot be estimated consistently. Instead, an approximation of the model, defined by the alternative hypothesis, can be used when testing nonlinearity (see [5, 6]). Examining the form of the logistic transition function, it is clear that if \(\gamma = 0\), then \(y_{i}=\textbf{x}_{i}^{\prime }\varvec{\beta }+G(s_i; \gamma , c)\textbf{x}_{i}^{\prime }\varvec{\psi }\) becomes linear. In addition, adding the assumption that \(\varvec{\psi }\ne \varvec{0}\), it is linear if and only if \(\gamma = 0\), and due to this fact, \(\gamma = 0\) can represent the linearity hypothesis [7].

To test the null of linearity, \(y_i = f\left( \gamma \right) \) is expanded by a first-order Taylor series expansion around \(\gamma = 0\). That is, the function \(G(s_i)\) is approximated locally around the null hypothesis and replaced in Eq. (9), such that

where \(\varvec{\phi }_1 = \left( \varvec{\beta }+\frac{\varvec{\psi }}{2}-\frac{\varvec{\psi }\gamma c}{4} \right) \) and \(\varvec{\phi }_2 = \frac{\varvec{\psi }\gamma }{4}\). Since \(\varvec{\phi }_2 = \varvec{0}\) if and only if \(\gamma = 0 \), the null hypothesis can be tested by the new null hypothesis \(H_0: \varvec{\phi }_2 = \varvec{0}\), where \(\varvec{\phi }_2\) is of dimension \((1 \times q)\), using the equation

This is a linear hypothesis in a linear model and can therefore be tested using standard asymptotic theory. Since the remainder, \(R_i\), is zero for all i under the null of linearity, the term \(e^{R_i} = 1\), and the asymptotic inference is not affected by the remainder when the null hypothesis is valid. The auxiliary model used for testing linearity can now be defined as

It is clear that

and the score of the auxiliary model under \(H_{0}\) is

where \(\hat{\alpha }\) is the ML estimator of \(\alpha \) and \(\hat{y}_{i}^{A}\) is calculated with the ML estimator \(\hat{\varvec{\theta }}\) under the null hypothesis of linearity. The observed outer-product estimator of the covariance matrix is then derived. The partitioned covariance matrix can be defined as

where

To define the LM test statistic, we reformulate the covariance matrix as

where \(\textbf{D}_{11i}\) denotes the \(((p+1) \times (p+1))\) upper left corner matrix of \(\textbf{D}_{i|H_{0}} \), \(\textbf{D}_{12i}\) the upper right \(((p+1) \times p)\) matrix where \(\textbf{D}_{12i} = \textbf{D}_{12i}^{\prime }\) and \(\textbf{D}_{22i}\) denotes the \((p \times p)\) lower right corner matrix of \(\textbf{L}_{22i}\). Note that p is the dimension of \(\varvec{\psi }\), i.e. the number of non-linear components to be tested.

Letting \(\textbf{D}_{(n)}\) denote the average

and using the inverse of a partitioned matrix

the LM statistic can be specified as

where \(\frac{\partial l_{i}\left( \alpha ,\varvec{\theta }\right) }{\partial \varvec{\phi }_{2}^{\prime }}\) is defined above, \(\textbf{D}_{ij} = \text {plim}_{n \rightarrow \infty }\textbf{D}_{ij \left( n\right) }\) with \(i,j = 1,2\) and \(\textbf{F}= \text {plim}_{n \rightarrow \infty } \textbf{F}_n\). Assume:

A6. The auxiliary matrix

exists and is positive definite with the logistic transition function \(G(r; \gamma ,c)\) where \(r \in (0,1)\). Then, under the null hypothesis of a linear model, the LM statistic can be proven to have a \(\chi ^2\) distribution with q degrees of freedom, see [7].

If the test result indicates the existence of a smooth transition, it is of interest to test for an additional one, see, e.g., [4, p. 384] or Appendix A.4 in [7]. Such a test is outlined below. In the case of two transitions, the hazard rate can be formulated as

with each of the two transition functions defined as above.

A test of the null that a second transition is present is an extension of the linearity test, where the parameter \(\gamma _2\) is now tested. That is, the first-order Taylor series expansion around \(\gamma _2 = 0\) derived as

which yields

where \(\varvec{\phi }_1 = \left( \varvec{\beta }+\frac{\varvec{\psi }_2 }{2}-\frac{\varvec{\psi }_2 \gamma _2 c_2}{4} \right) \) and \(\varvec{\phi }_2 = \frac{\varvec{\psi }_2 \gamma _2 }{4}\). Since \(\varvec{\phi }_2 = \varvec{0}\) if and only if \(\gamma _2 = 0 \), the null hypothesis of only one transition can be tested by the new null hypothesis \(H_0: \varvec{\phi }_2 = \varvec{0}\). As before, the ith element of the estimated covariance matrix of the score under the null can be written as

where now

and

The test is then equivalent to the one presented above.

The performance of the LM-test of linearity is examined with a simulation experiment where data are generated using a duration model with an intercept and two variables where

First, a test of a nonlinear adjustment involving only the intercept, that is, the null of \(\phi _{20} = 0\) is examined. Second, a test of whether the intercept and impact of control variables are changing, i.e. \(\varvec{\phi }_2 = 0\), is examined. The results regarding the size of the test for six different sample sizes are presented in Table \(1\). The empirical size of the linearity test seems close to the nominal size, even if there are some differences for the smaller sample sizes. One can especially note that the empirical size is slightly worse when testing \(\varvec{\phi }_2 = 0\) compared to \(\phi _{20} = 0\).

The size-adjusted power of the test is presented in Table \(2\). The parameters under the alternative are \( \varvec{\psi }= (0.25, 0.25, 0.25)\) and K is the number of parameters tested, only the intercept or both the intercept and impact of control variables. Results where \( \varvec{\psi }= (0.25, 0.25, 0.25)\) is scaled with 0.5 is presented in Table \(5\) in Appendix. The results are as expected, with the power of the tests approaching one for large sample sizes and where both the scaling and value of the slope parameter affect the power. We can, in addition, as for the empirical size, note that the test performs slightly better when testing only one parameter compared to testing three. The empirical implication is that one should not routinely test for nonlinearity using all variables but select a smaller subset, perhaps only the constant, based on some sound thematic theory.

Electricity markets in Queensland have gradually moved from a regulated system towards a more competitive market structure within the Australian National Electricity Market (NEM).

During the 2000s a series of reforms aimed to increase competition in electricity retailing, culminating in the introduction of full retail contestability in 2007, which allowed consumers to choose among competing electricity retailers. As competition expanded, multiple retailers entered the market and offered a range of contracts with different pricing structures (see e.g. [17]). These developments reflect the broader deregulation process in the electricity sector during the sample period.

According to the Department of Foreign Affairs and Trade [18, 19], electricity prices experienced greater volatility and upward pressure after the mid-2000s, particularly following policy uncertainties around 2007, and retail prices showed substantial dispersion as competition expanded. Such institutional changes may influence the dynamics of electricity prices and the occurrence of extreme price events. In particular, changes in market structure and competitive conditions may affect the frequency and timing of abnormal price spikes, which in a duration framework would manifest as changes in the waiting time between events.

A reason for deregulating is to make the energy markets more efficient as argued by, e.g., [20, 21]. The deregulation has been studied previously by, e.g., [17]. They find that the deregulation has increased competition and led to greater price dispersion. A more efficient market would imply that abnormal price events would occur more rarely. Abnormal price events have been observed in various electricity price markets, see, e.g., [22,23,24]. The Australian Energy Regulator is required to publish reports whenever the price exceeds 5,000 Australian dollars per MWh occur. In the Australian context, abnormal price events have been documented and analysed by [25,26,27]. The relationship between abnormal price events and price deregulation in Queensland, Australia has been analysed by [7]. They conclude that the competitive environment has been accompanied by a increased strategic behaviour. This is demonstrated by an increased probability of abnormal price events. Details of the Australian energy market can be found at the Australian Energy Regulator, www.aer.gov.au.

The motivation for the use of the duration model outlined in this paper is to analyse if strategic price behaviour is more frequent after deregulation. To do this, we model the duration of the denoted price episodes in the Queensland electricity market. A price episode is defined as the duration of time between two price events, measured in half hours. A price event is defined as one or multiple consecutive half hours with a spot price above an inflation-adjusted price corresponding to \(\$80 \) in 2001. As stated in [7], this figure is chosen to reflect the marginal cost of electricity generation by retailers in the Queensland region. This way, it is possible to assess the impact of deregulation by examining whether or not the time until the occurrence of an episode has decreased after the changed market conditions.

A closer look at how electricity is physically traded between generators and market consumers is needed to understand the possible strategic behaviour in the electricity market. Ahead of each trading day, the market generators, who sell output through the spot market and receive the spot price at settlement, provide details on their availability and offer to produce particular quantities at particular prices. The bids indicate how many megawatts the generator wishes to produce in up to 10 price bands at particular prices. Generators can bid up to a price cap of \(\$10{,}000\) per MWh, and the market floor price is minus \(\$1000\) per MWh. These bids also include information about the generators’ minimum operating level, where the price bid changes from a negative to a positive price band. This is a level at which the generator can operate indefinitely without requiring auxiliary firing to respond to changed dispatch instructions. The generators can change the bid quantities until 5 min before actual dispatch. Trading in a specific region of the National Electricity Market (NEM) is then based on a 30-min trading interval, where the spot price is the average of the six 5-min interval dispatch price outcomes for the preceding half hour.

Since the average spot price of a half-hour interval is inflated by an abnormally high dispatch price recorded for any 5-min interval, one can argue that there are incentives for strategic behaviour in the bidding process. The speed at which generating capacity can be increased is generally a lot faster than the time it takes for a generator not currently dispatched to start injecting electricity. This implies that generators already dispatched, but not at full capacity, in previous intervals are the better choice, giving them a strategic advantage. One scenario could be that base load generators withhold capacity at the lowest price bands as the dispatched load approaches a critical point in terms of capacity. If the existing bids from the base load generators are fully dispatched and load is still required, the market operator must dispatch generators who have bid at higher prices. Once the price is forced up, the base load generators can rebid all their available capacity in the subsequent 5-min intervals. The half-hour spot price is thereby forced up.

The dataset comprises half-hour data from January 1, 2001, to June 28, 2018, a total of 301,134 observations. The variable of interest is the half-hour spot electricity price per MWh in Queensland, Australia. Of particular interest are the cases where the spot price is above a \(\$80 \) for one or many consecutive time points. Such observations are denoted price events. If the price is above \(\$80\) for exactly one half-hour, the observation is denoted an abnormal price event.

Hurn et al. [7] aim to find out whether the number of longer-lasting price episodes has decreased or not since 2007. We investigate another closely related question: whether the number of consecutive half hours between two price events has decreased over time.

As mentioned in [7], the standard explanation for the occurrence of price events might follow increased demand due to, for example, extreme weather conditions and decreasing supply due to generation failure. Price episodes can therefore be explained by scarcity and do not generally imply any strategic behaviour by market participants. The abnormal price events, on the other hand, might be caused by incentives to increase the spot price.

There are 2985 price events registered in the study period and 1558 of these are abnormal price events. The number of price events per six months is presented in Fig. 1a. There is an increased number of events in the first half of 2007, when the deregulation started, followed by a decrease. There seems to be an indication that abnormal price events are occurring more often in recent years. The proportion of all price events that lasted for precisely one half-hour was \(40\%\) in 2001 and increased to \(63\%\) in 2017, see Fig. 1b. The main interest is to examine if the duration until a price event is affected by deregulation and the move towards full retail competition in the electricity market, in the sense that abnormal price events might be allowed to occur more often.

Temperature effects can be assumed to be important in periods of abnormal electricity spot prices. As in [7], three measures of temperature are included as control variables in the analysis of time between price events. First, the daily temperature range (\( TR_i \)) on day \(t_i\) is computed. In addition, two variables capturing extreme weather conditions are constructed based on heating degree days and cooling degree days. These are computed as

where \(d(t_i)\) is the day of the ith price event, TT is a threshold temperature, in this case set to \(18\,^{\circ }\)C, and \(\bar{T}_d\) is the average temperature on the day d. \(K = 3\) is used.

The inter-connector flow between the region of New South Wales and Queensland (QNI) is another important variable when examining price effects. If the QNI is positive, the flow is north into Queensland from New South Wales. A positive flow indicates that the Queensland system is under some stress, which could lead to longer-lasting episodes with increased prices. Also, an unexpected load is likely associated with an increase in prices. Therefore, the difference between the actual load and an estimate of the load from an autoregressive model is included as a measure of unexpected demand (UL). To account for potential duration dependence across spells of price events, the length of the preceding spell (last event, LE) is included as a covariate. Summary statistics for the covariates are presented in Table 3.

The results from the estimation of an ordinary Weibull survival model, examining the duration model when \(y_i\) is linear, are presented in the first column of Table 4. The estimated baseline hazard is less than one, indicating a decreased hazard rate with duration time. The effect of both the cumulative number of heating and cooling degree days is positive, as expected. The daily temperature range is found to have a weak negative impact on the rate at which price events occur. Since the difference between the daily maximum and minimum temperatures is smaller in the summer, this could reflect that the system operates much closer to capacity than during winter. The effect of the unexpected load is positive. The higher the unexpected load, the higher the rate at which price events occur. The same is true for QNI, as a positive flow indicates some stress to the Queensland system. Finally, the length of the last event seems to positively affect the rate of price event occurrences.

The null hypothesis of linearity is tested against an alternative of a nonlinear model with one transition. With a resulting LM-type statistic of \(322 > \chi ^2_{7, 0.05} = 14\), the null is rejected. The full nonlinear smooth transition duration model is then estimated. First, a grid search determines the starting values to avoid getting stuck on local minima. A search over a set of fixed parameters \((\eta , c)\) is performed, and the set maximizing the log-likelihood is chosen as starting values. The result of the full nonlinear model with one transition (i) is presented in the second column of Table 4.

All parameter estimates of the covariates in the linear part have the same sign and are approximately unchanged, except for the temperature range now being non-significant. The intercept estimate is positive, indicating that the rate at which price events occur seems to increase with the transition, holding all covariates constant. The result of the contribution of cooling degree days, QNI, and the last event to the hazard function also indicates a significant change over the years. We perform an LR test, and the null hypothesis that only the intercept is changing with the transition against the alternative of a full nonlinear model is rejected. The estimated mid-point of the transition is at \(c=0.86\), that is, in January 2016. The slope parameter estimate is \(\eta = 5.38\), corresponding to a smooth transition over about a year, from August 2015 until June 2016. This coincides with the deregulation of the retail energy prices in South East Queensland which occurred at first of July 2016. This would indicate a expectation driven change in behaviour.

A formal test of the null of a single transition against the alternative of two transitions is conducted. With a test statistic of \(169 > \chi ^2_{7, 0.05} = 14\), the null of only one transition is rejected. An intercept only (ii) and a full two-transition model (iii) are therefore estimated, and the results are presented in the third and fourth columns of Table 4.

With \(c_1=0.34\), the second transition seems to occur between August 2006 and June 2007, a few months before the deregulation. The hypothesis that the impact of all the covariates does not change with the second transition cannot be rejected, but only the last event variable is significant. Comparing log-likelihoods, AIC, and BIC for the models with two transitions, see Table 6 in Appendix. Model (ii) seems the most relevant in capturing the changing market conditions, with an increase in the rate at which price events occur at the beginning of 2007. There are, at least, two possible explanations for the transition being slightly early. The first is that the result is due to estimation uncertainty. It is widely known that estimating the exact timing is rather difficult in smooth transition models, [6]. The other is that there might be an expectation component where the market starts to adjust prior. There is yet another possible reason. According to [28] there were political risks due to the Kyoto protocol (increasing the volatility) meanwhile generators reaching end of life time. These are temporary effects on prices but can explain that the timing is prior to the deregulation in 2007.

When continuing with a test of a third transition, the null can again be rejected. The result of a three transition model (iv) is presented in the fifth column of Table 4. The estimation results indicate a third transition with a midpoint in August 2007, where the rate at which price events occur now seems to decrease. This result could indicate a move back to market conditions before the deregulations. As the timing of the third transition is just slightly behind the second but with less slope the total effect is first an increased effect which then damps of (due to the negative constant in the third transition). Overall, this would imply an initial stronger reaction which then returns back towards the situation before August 2006.

The three transition functions from this model are presented in Fig. 2a–c.

The survival function for model (iv) is presented in Fig. 3, where the different regimes are compared. The probability of a duration to hold until a certain time point decreases with the transition to the second regime in late 2006, with a succeeding increase in 2007. After 2015, the probability decreases again when all transitions in model (iv) have occurred.

A qq-plot is displayed in Fig. 4 where the standardized residuals (\((\ln t - \hat{\mu } - \hat{\varvec{\beta }}^T\textbf{x})/\hat{\sigma }\)) are plotted versus a theoretical standard extreme value distribution. For a motivation, see e.g. [29, Page 415]. As in the covariates are time dependent, the qq-plot is based on the covariates at the time of event. This implies that some caution needs to be taken when interpreting the qq-plot. The plot includes 95-% confidence bands simulated using the theoretical distribution. It can be seen that the distribution has subpar fit in the very left tail, appropriate fit in the major middle part of the distribution. Although it bends of in the upper tail it is in the boarder of the confidence band. Overall, the qq-plot indicates that the Weibull assumptions yields an adequate fit.

To examine a smooth versus an abrupt change empirically, we compare the smooth transition model with models with a deterministic structural break included sequentially for each month, instead of a smooth transition, see Fig. 5a–c. This serves as a robustness check for the location of the change point [3, 7]. The left axis and the solid line represent the log-likelihood of the deterministic structural break models, and the dotted line (right axis) is the specific smooth transition function found by estimating model (iv) in Table 4. Figure 5a represents the log-likelihood from estimating a linear model with a dummy variable being zero until the specified month and one thereafter for each month between January 2002 to July 2017. Figure 5b and c corresponds to the log-likelihood from models including a deterministic structural break in January 2016 and both January 2016 and July 2007, respectively, and with an additional break included as described above.

To some extent, the breaks maximizing the log-likelihood of the linear model agree with the estimated transitions. However, the second transition function indicates a transition somewhat before the linear model with a maximized log-likelihood given a structural break in April 2007. The estimation results from the three linear models with deterministic structural breaks maximizing the log-likelihoods are presented in Table 7. The estimated parameters show an increase in the hazard at the first two time points and a decrease in the third, with about the same magnitudes as in the smooth transition models.

It is of interest to compare our results with those reported in [7], particularly with respect to the transition periods. The primary difference is that we detect three transitions, whereas they report only one. Specifically, their estimated transition occurs between August 2008 and July 2009, while our estimated transitions are found in August 2015 until June 2016, August 2006 until June 2007, and April 2007 until January 2008, respectively. Note that our first is outside their data range. As the Weibull model is less restrictive than the logistic it can adapt better to the data and the statistical tests be more powerful. Hence, one interpretation is that they only found one transition due to low power of their test. An additional reason might be that our sample is larger which also increase power, especially in the end of the sample.

In all of the above specifications, the power dependence of the hazard is assumed and estimated to be less than one. This implies a decreased hazard rate with duration time. As a check of the assumption of power dependence, a comparison with a constant baseline hazard is examined. However, it seems adequate to assume an, in this case, decreasing power dependence.

In this paper, we introduce a smooth transition duration model, allowing time-varying covariates and duration time to vary with smooth transitions over different regimes. Using Monte Carlo simulations, the proposed LM tests are shown to have the desired properties.

In our empirical application, we model time until an unexplained, short price increase, i.e., an abnormal price event, in the electricity spot price in Queensland, Australia. By modelling this duration time, we investigate if there is a change over time and if, more specifically, the behaviour of market participants is affected by regulatory changes. The results show that there seems to be more than one transition in the study period. The first transition is found to take place at the beginning of 2016, while the second is found in the first half of 2007, indicating an increase in the rate at which price events occur. As the Australian electricity market deregulation began in 2007, with energy prices in South East Queensland deregulated in 2016, the findings are in line with what could be expected. A third transition, with a midpoint in August 2007, can indicate a move back to market conditions before the deregulations.

The structural changes seem, in addition, not to be instantaneous but rather smooth over a period of time, justifying the use of the proposed model. The results are in line with the conclusions of [7], who find support for the hypothesis that the deregulation altered the behaviour of market participants. This has important policy implications as previous research (e.g. [17, 20, 21]) have found that the deregulated energy markets improves efficiency while our results implies that an unregulated market needs monitoring. The monitoring should have the aim of distinguish between abnormal price events due to market manipulations and those due to natural reasons such as unexpected changes in supply and demand.